Chapter 15

Interest-Rate Modelling and Derivative Pricing

15.1 Basic Fixed Income Instruments

15.1.1 Bonds

The term fixed income refers to any type of investment that provides payments of a fixed amount on a fixed schedule. A most common type of fixed income investment is a bond. In Chapter 1, we introduced a zero-coupon bond (ZCB) that only returns the investor a redemption amount on the maturity date. Another example of a fixed income security is a coupon bond that provides regular payments (coupons) on a fixed schedule and a redemption value on the maturity date. Recall that Z(t, T) denotes the time-t purchase price of a unit zero-coupon bond maturing at time T with 0 ≤ t ≤ T, whose face value is equal to $1. The following notations were also introduced in Chapter 1.

y is the continuously compounded yield rate (also called the spot rate); it is generally a function of calendar time t and maturity T for a given investment time period [t, T], i.e., y = y(t, T). When regarded as a function of the time to maturity, τ = T − t, i.e., y(τ) is the yield rate earned by money invested for a period of τ years.

r is the short rate; it is a function of calendar time t, i.e., r = r(t). The short rate is the rate on instantaneous borrowing or lending and is defined as r(t)= y(t, t).

The continuously compounded yield rate y and zero-coupon bond price Z are simply related by

$Z\left(t,T\right)={\text{e}}^{-y\left(t,T\right)\left(T-t\right)}.\text{ (15.1)}$

Any cash flow stream of multiple coupon payments can be replicated by means of a portfolio of zero-coupon bonds. In particular, the time-t value V(t; c, T) of a cash flow stream (c, T) with c =[c1, c2, ..., cn] and T =[t1, t2, ..., tn], where ci is the payment at time ti and t ≤ t1 < t2 < ··· < tn holds, is equal to the sum of discounted cash flows,

$V\left(t;c,T\right)={\displaystyle \sum _{i=1}^n{c}_i}Z\left(t,t_i\right)={\displaystyle \sum _{i=1}^n{c}_i}{\text{e}}^{-y\left(t,t_i\right)\left(t_i-t\right)}.\text{ (15.2)}$

The above formula allows for pricing coupon-paying bonds. On the other hand, if we are given the price of a coupon-paying bond for each maturity t1, t2, ..., tn, then using (15.2) we can solve recursively for the prices Z(t, t1), Z(t, t2), ..., Z(t, tn) and therefore obtain the yield rates y(t, t1), y(t, t2), ..., y(t, tn). Indeed, let the coupon bond value Vi(t) = c1Z(t, t1) + c2Z(t, t2) + ··· + ciZ(t, ti) be known for all i = 1, 2, ..., n. Then the ZCB values are

$Z\left(t,t_1\right)=\frac{V_1\left(t\right)}{c_1}\text{and}Z\left(t,t_i\right)=\frac{V_i\left(t\right)-V_{i-1}\left(t\right)}{c_i}\text{for}i=2,3,...,n.$

This method of deducing zero-coupon bond values from coupon-paying bond prices is called bootstrapping.

15.1.2 Forward Rates

Suppose we wish to borrow an amount of A dollars for a period between times T and T′ and we want to lock in the rate on the loan at time t with t ≤ T < T′. To do this, at time t, we purchase A units of the zero-coupon (unit) bond maturing at time T and finance this purchase by selling short $A\frac{Z\left(t,T\right)}{Z\left(t,T^{\prime }\right)}$ units of a zero-coupon bond maturing at time T′. The cost of setting up this portfolio is zero. At time T, we receive $A from the long position in the T-maturity bonds. At time T′, we are required to have $A\frac{Z\left(t,T\right)}{Z\left(t,T^{\prime }\right)}$ dollars to cover the short position in the T′-maturity bonds. Hence, to avoid arbitrage, A dollars invested during the time interval [T, T′] must yield an effective (continuously compounded) rate, denoted by f(t; T, T′), such that

$A{\text{e}}^{\left(T^{\prime }-T\right)f\left(t;T,T^{\prime }\right)}=A\frac{Z\left(t,T\right)}{Z\left(t,T^{\prime }\right)}\Rightarrow f\left(t;T,T^{\prime }\right)=\frac{1}{T^{\prime }-T}\ln \frac{Z\left(t,T\right)}{Z\left(t,T^{\prime }\right)}.\text{ (15.3)}$

The rate f(t; T, T′) is called a forward rate. It is determined at time t for investing during the period [T, T′]. Using (15.1), we express the forward rate in terms of yield rates as follows:

$f\left(t;T,T^{\prime }\right)=\frac{1}{T^{\prime }-T}\ln \left(\frac{{\text{e}}^{-y\left(t,T\right)\left(T,-t\right)}}{{\text{e}}^{-y\left(t,T^{\prime }\right)\left(T^{\prime },-t\right)}}\right)=y\left(t,T^{\prime }\right)\frac{T^{\prime }-t}{T^{\prime }-T}-y\left(t,T\right)\frac{T-t}{T^{\prime }-T}.\text{ (15.4)}$

Note that, when t = T the forward rate is determined in the beginning of the investment interval and we have

$f\left(T;T,T^{\prime }\right)=y\left(T,T^{\prime }\right)\frac{T^{\prime }-T}{T^{\prime }-T}-y\left(T,T\right)\frac{T-T}{T^{\prime }-T}=y\left(T-T^{\prime }\right).$

That is, yield rates are forward rates for immediate delivery.

The instantaneous forward rate of maturity T is defined as

$f\left(t,T\right):=\underset{T^{\prime }\to T}{\lim }f\left(t;T,T^{\prime }\right).\text{ (15.5)}$

By applying the definition of the derivative of ln Z(t, T) w.r.t. T, for fixed t, (15.3) gives the instantaneous forward rate at time T as

$f\left(t,T\right):=\underset{T^{\prime }\to T}{\lim }f\left(t;T,T^{\prime }\right)=-\underset{T^{\prime }\to T}{\lim }\frac{\ln Z\left(t,T^{\prime }\right)-\ln Z\left(t,T\right)}{T^{\prime }-T}=-\frac{\partial }{\partial T}\ln Z\left(t,T\right).\text{ (15.6)}$

Hence, integrating this equation, where $f\left(t,u\right)=-\frac{\partial }{\partial u}$ ln Z(t, u), for u ∊ [t, T], while using Z(t, t) = 1, gives the zero-coupon bond price in terms of instantaneous forward rates,

$Z\left(t,T\right)=\exp \left(-{\displaystyle {\int }_t^{T}f\left(t,u\right)\text{d}u}\right).\text{ (15.7)}$

Taking the logarithm of both sides of (15.7) gives

${\displaystyle {\int }_t^{T}f\left(t,u\right)\text{d}u=-\ln Z\left(t,T\right).}\text{ (15.8)}$

Using (15.8), we can find the integral of the instantaneous forward rate of maturity changing from T to T′ with t ≤ T ≤ T′:

$\begin{array}{c}{\displaystyle {\int }_T^{T^{\prime }}f\left(t,u\right)\text{d}u}={\displaystyle {\int }_t^{T^{\prime }}f\left(t,u\right)\text{d}u-{\displaystyle {\int }_t^{T^{\prime }}f\left(t,u\right)\text{d}u}=\ln Z\left(t,T\right)-\ln Z\left(t,T^{\prime }\right)}\\ =f\left(t;T,T^{\prime }\right)\left(T^{\prime }-T\right).\end{array}\text{ (15.9)}$

The forward rate is also related to the forward price for a unit zero-coupon bond maturing at time T′ with settlement at time T. Recall that a forward contract is a contract under which one party is obliged to sell a specified asset for an agreed price to the other party at a designated future date. A fixed income forward contract is an agreement between two parties to pay a specified delivery price for a fixed income security (a zero-coupon bond, for instance) at a given delivery date. The forward price of the underlying security is the value of the delivery price that makes the forward contract have no-arbitrage price zero at initiation.

15.1.3 Arbitrage-Free Pricing

The risk-free interest rate has been assumed to be a constant or a deterministic function for most of the option pricing models applied in previous chapters. In this chapter, we will deal with stochastic models of interest rates. Assume a filtered probability space (Ω, ℱ, ℙ, $\mathbb{F}$), where $\mathbb{F}={\left\{{ℱ}_t\right\}}_{0\le t\le T*}$ is a filtration generated by the stochastic risk-free (short) rate process ${\left\{r\left(t\right)\right\}}_{0\le t\le T*}$ for some T* > 0. Our general assumptions are as follows:

1. the short rate process is Markovian;
2. the zero-coupon bond price process {Z(t, T)}0≤t≤T is adapted to $\mathbb{F}$ for every maturity T with T ≤ T*

As in previous chapters, the bank account {B(t)}t≥0 evolves according to

$dB\left(t\right)=r\left(t\right)B\left(t\right)\text{d}t,B\left(0\right)=1,\text{ (15.10)}$

and is given by

$B\left(t\right)=\exp \left({\displaystyle {\int }_0^{t}r\left(s\right)}\text{d}s\right).$

Since B(t) is a function of short rates {r(s)}0≤s≤t, the bank account is adapted to the filtration F. Additionally, let us define the (stochastic) discount factor D(t, T) from time t to time T with 0 ≤ t ≤ T ≤ T* given by

$D\left(t,T\right)=\frac{B\left(t\right)}{B\left(T\right)}=\exp \left(-{\displaystyle {\int }_t^{T}r\left(s\right)}\text{d}s\right).$

The discount factor has the property D(t, T)D(T, T′) = D(t, T′) for all 0 ≤ t ≤ T ≤ T′. Also, we recall from previous chapters that D(t) := D(0, t) = B−1(t) for t ≤ 0.

One of the main problems discussed in this chapter is the no-arbitrage pricing of bonds, options on interest rate, and other fixed income derivatives. Here, bonds can be regarded as derivative assets since any bond is derived from the knowledge of the short (risk-free) rate, r(t), which takes on the role of the underlying. The Fundamental Theorem of Asset Pricing (FTAP) is a cornerstone of the no-arbitrage pricing. Let us state the FTAP for the model of stochastic interest rates.

Theorem 15.1.

The market is arbitrage free if there exists a probability measure $\tilde{\mathbb{P}}$ equivalent to the real-world measure ℙ, under which the discounted zero-coupon bond price

$\overline{Z}\left(t,T\right):=Z\left(t,T\right)D\left(t\right)=Z\left(t,T\right)/B\left(t\right),0\le t\le T,$

is a martingale for each T > 0. Assuming the absence of arbitrage, the market is complete iff the equivalent martingale measure $\tilde{\mathbb{P}}$ is unique.

The definition of a zero-coupon bond and definition of a martingale imply that1

$\overline{Z}\left(t,T\right)={\tilde{\text{E}}}_t\left[\overline{Z}\left(T,T\right)\right]={\tilde{\text{E}}}_t\left[Z\left(T,T\right)D\left(T\right)\right]={\tilde{\text{E}}}_t\left[D\left(T\right)\right].$

Thus, the time-t price of the zero-coupon bond is

$Z\left(t,T\right)={\tilde{\text{E}}}_t\left[B\left(t\right)D\left(T\right)\right]={\tilde{\text{E}}}_t\left[D\left(t,T\right)\right]={\tilde{\text{E}}}_t\left[\exp \left(-{\displaystyle {\int }_t^{T}r\left(s\right)\text{d}s}\right)\right].\text{ (15.11)}$

By using the results of the FTAP, we can also price derivatives. The time-t no-arbitrage value V(t) of a payoff V(T) payable at time T with 0 ≤ t ≤ T (note that V(T) is ℱT-measurable) is

$V\left(t\right)={\tilde{\text{E}}}_t\left[D\left(t,T\right)V\left(T\right)\right]={\tilde{\text{E}}}_t\left[\exp \left(-{\displaystyle {\int }_t^{T}r\left(s\right)\text{d}s}\right)V\left(T\right)\right].\text{ (15.12)}$

As an example, consider a forward contract under which $K will be paid at time T in return for a repayment of $1 at time T′ with T < T′. Equivalently, this contract is arranged as if a zero-coupon bond maturing at time T′ is delivered at time T in return for K dollars paid at the same time T. The time-T payoff is Z(T, T′) − K. According to (15.12), the price of the forward contract at time t (with t ≤ T) is

$\begin{array}{c}V\left(t\right)={\tilde{\text{E}}}_t\left[D\left(t,T\right)\left(Z\left(T,T^{\prime }\right)-K\right)\right]={\tilde{\text{E}}}_t\left[D\left(t,T\right)\left({\tilde{\text{E}}}_t\left[D\left(T,T^{\prime }\right)\right]-K\right)\right]\\ ={\tilde{\text{E}}}_t\left[D\left(t,T^{\prime }\right)\right]-K{\tilde{\text{E}}}_t\left[D\left(t,T\right)\right]=Z\left(t,T^{\prime }\right)-KZ\left(t,T\right).\end{array}$

Here, we combined the tower property with $Z\left(T,T^{\prime }\right)={\tilde{\text{E}}}_T\left[D\left(T,T^{\prime }\right)\right]$ and the identity D(t, T)D(T, T′) = D(t, T′). Choosing K = Z(t, T′)/Z(t, T) ensures that V(t) = 0. This value is called the T-forward price of the zero-coupon bond with maturity T′ > T. The forward price is expressed in terms of the forward rate f(t; T, T′) as given in (15.3).

Consider now an asset X with price process {X(t)}0≤t≤T and a forward contract that delivers one unit of X at time T in return for K. According to (15.12), the time-t price of this contract is

$V\left(t\right)={\tilde{\text{E}}}_t\left[D\left(t,T\right)\left(X\left(T\right)-K\right)\right]=\frac{1}{D\left(t\right)}{\tilde{\text{E}}}_t\left[D\left(T\right)X\left(T\right)\right]-K{\tilde{\text{E}}}_t\left[D\left(t,T\right)\right].$

Since the discounted process D(t)X(t) is a $\tilde{\mathbb{P}}$-martingale and ${\tilde{\text{E}}}_t\left[D\left(t,T\right)\right]=Z\left(t,T\right)$, the above equation reduces to

$V\left(t\right)=X\left(t\right)-KZ\left(t,T\right).$

The present (time-t) value is zero iff $K=\frac{X\left(t\right)}{Z\left(t,T\right)}$. We call $\frac{X\left(t\right)}{Z\left(t,T\right)}$ the T-forward price at time t ∊ [0, T] of the asset X. Note that X(t) = Z(t, T′) is a special case where the chosen asset is the time-T′ maturity zero-coupon bond.

15.1.4 Fixed Income Derivatives

15.1.4.1 Options on Bonds

A European-style option on a zero-coupon bond is defined in the same way as the option on any other underlying security. A call option with maturity T and strike K on a bond maturing at time T′ > T gives its holder the right but not the obligation to purchase the bond at time T for a fixed price K. The time-T payoff to the call option holder is (Z(T, T′) − K)+. Similarly, we define a put option with the time-T payoff (K − Z(T, T′))+ . If the joint (conditional) distribution of the discount factor D(t, T) and bond price Z(T, T′) is known, then the no-arbitrage price at time t of a European option with payoff V(T) = Λ(Z(T, T′)) can be calculated using the risk-neutral pricing formula (15.12):

$V\left(t\right)={\tilde{\text{E}}}_t\left[D\left(t,T\right)\Lambda \left(Z\left(T,T^{\prime }\right)\right)\right],0\le t\le T.$

Note that many other common options on interest rates can be expressed as options on bonds. For example, a call option on the LIBOR rate considered in the last section of this chapter is equivalent to a put option on a zero-coupon bond.

We can also consider a European call option written on a coupon-bearing bond. The payoff of the option struck at exercise K, of maturity date T, can be written as V(T) = (P (T) − K)+, where P (T) is the value of the bond at maturity T :

$P\left(T\right)={\displaystyle \sum _{j=1}^n{c}_j}Z\left(T,T_j\right),$

with cash flows c1, c2, ..., cn at times T1, T2, ..., Tn, respectively, with T < T1 < T2 < ... < Tn. Note that the sum involves all cash flows at future times past the maturity of the option, discounted back to time T.

15.1.4.2 Cap and Caplets

A caplet is a contract that gives its holder the right to pay the smaller of two simple interest rates: the floating rate f and fixed rate κ. The floating rate is typically the three-or six-month LIBOR. For the holder of a caplet over the interval [T, T + τ] with tenor τ, the rate of payment is capped at κ from T to T + τ. So the simple interest paid on each dollar of the principal at time T + τ is the smaller of τf and τκ. Since without the caplet the interest payment would be τf, the caplet’s worth to the holder is τf − min{τf, τκ} =(f − κ)+τ . That is, the caplet pays (f − κ)+τ to its holder at time T + τ . In fact, we deal with a call payoff on the floating rate f with strike κ and maturity T + τ . Under the equivalent martingale measure (EMM) $\tilde{\mathbb{P}}$, the time-t value of the caplet is

${\text{Caplet}}_{T+\tau }\left(t\right)=\tau {\tilde{\text{E}}}_t\left[D\left(t,T+\tau \right){\left(f-k\right)}^{+}\right],0\le t\le T.\text{ (15.13)}$

As will be demonstrated below, the LIBOR rate is expressed in terms of zero-coupon bonds, and hence the application of the risk-neutral pricing formula in (15.13) is legitimate.

A cap is defined as a collection of caplets. Suppose that the payments are done at the times T1, T2, ..., Tn with Ti + 1 = Ti + τ. Let fi denote the floating rate over [Ti−1, Ti] for i = 1, 2, ..., n. A cap is defined as a stream of cash flows that pays to its holder (fi − κ)+τ at time Ti for all i = 1, 2, ..., n. The risk-neutral value of the cap with the tenor structure T := [T1, T2, ..., Tn] at time t ≤ T0 is

$\text{Cap}\left(t,T\right)={\displaystyle \sum _{i=1}^n{\text{Caplet}}_{T_i}\left(t\right)}=\tau {\tilde{\text{E}}}_t\left[{\displaystyle \sum _{i=1}^{n}D\left(t,T_i\right){\left(f_i-k\right)}^{+}}\right].\text{ (15.14)}$

15.1.4.3 Swap and Swaptions

Another basic fixed income instrument is an interest rate swap. It is an agreement between two parties in which one party makes fixed interest payments on some notional amount at regularly spaced dates in return for floating interest payments on the same principal at the same dates. The payer swap is a contract in which the floating rate fi is swapped in arrears against a fixed rate κ at n intervals [Ti−1, Ti] of length τ = Ti−Ti−1 for all i = 1, 2, ..., n. The holder of the payer swap receives the cash flows (f1 − κ)τ, . . . , (fn − κ)τ at dates T1, ..., Tn, respectively. The other party in the swap contract enters a receiver swap, in which a fixed rate is swapped against the floating rate. It suffices to only consider a payer swap which we simply call a swap. The time-t value of the swap is

$\text{Swap}\left(t;T\right)=\tau {\displaystyle \sum _{i=1}^n{\tilde{\text{E}}}_t}\left[D\left(t,T_i\right)\left(f_i-k\right)\right].\text{ (15.15)}$

The fixed rate κ can be defined so that the swap agreement costs zero at initiation. The forward swap rate is a fixed rate of interest that makes the swap contract worthless at current time t ≤ T0. In other words, the forward swap rate at time t is the rate κ that solves Swap(t; T) = 0.

Another contract called a swaption gives you the right but not the obligation to enter into a swap agreement with another party at time T0. So a swaption delivers at time T0 a swap when the swap value Swap(T0; T) is positive. Thus, the time-t value of a swaption at time t ≤ T0 is

$\text{Swaption}\left(t;T\right)={\tilde{\text{E}}}_t\left[D\left(t,T_0\right){\left(\text{Swap}\left(T_0;T\right)\right)}^{+}\right].\text{ (15.16)}$

15.2 Single-Factor Models

A single-factor model is one that has a single, one-dimensional source of randomness affecting bond prices. Let a standard Brownian motion {W (t)}t≥0 be such a source of randomness. Let the Brownian filtration ${\mathbb{F}}^W:={\left\{{ℱ}_t^W\right\}}_{0\le t\le T*}$ coincide with the filtration $\mathbb{F}$ generated by the short rate process ${\left\{r\left(t\right)\right\}}_{0\le t\le T*}$. This is the case when r(t) is an Itô process governed by a nontrivial stochastic differential equation (SDE) of the form dr(t) = a(t)dt + b(t)dW (t) with $\mathbb{F}$W-adapted processes a and b.

Let us derive an SDE for the zero-coupon bond Z(t, T) under the EMM $\tilde{\mathbb{P}}$. The discounted process $\overline{Z}\left(t,T\right):=D\left(t\right)Z\left(t,T\right)$ for 0 ≤ t ≤ T, and a given fixed T > 0, is a $\tilde{\mathbb{P}}$-martingale. By the Brownian Martingale Representation Theorem 11.14, there exists an $\mathbb{F}$-adapted process θ(t) such that $\overline{Z}\left(t,T\right)=\overline{Z}\left(0,T\right)+{\displaystyle {\int }_0^t\theta }\left(s\right)\text{d}\tilde{W}\left(s\right)$ (a.s.) for 0 ≤ t ≤ T. Since $\overline{Z}\left(t,T\right)$ is strictly positive, we can define ${\sigma }_Z\left(t,T\right)=\theta \left(t\right)/\overline{Z}\left(t,T\right)$. Thus, we have

$\text{d}\overline{Z}\left(t,T\right)=\theta \left(t\right)\text{d}\tilde{W}\left(t\right)=\overline{Z}\left(t,T\right){\sigma }_Z\left(t,T\right)\text{d}\tilde{W}\left(t\right).$

By using the Itô product rule, we obtain the following expression for the stochastic differential of the discounted bond price:

$\text{d}\overline{Z}\left(t,T\right)=-\overline{Z}\left(t,T\right)r\left(t\right)\text{d}t+D\left(t\right)\text{d}Z\left(t,T\right).$

As a result, we have the following SDE for Z(t, T) under the EMM $\tilde{\mathbb{P}}$:

$\frac{\text{d}Z\left(t,T\right)}{Z\left(t,T\right)}=r\left(t\right)\text{d}t+{\sigma }_Z\left(t,T\right)\text{d}\tilde{W}\left(t\right).\text{ (15.17)}$

This SDE has the equivalent integral form

$Z\left(t,T\right)=Z\left(0,T\right){\text{e}}^{f_0^{t}r\left(s\right)\text{d}s-\frac{1}{2}{f}_0^t{\sigma }_Z^2\left(s,T\right)\text{d}s+f_0^t{\sigma }_Z\left(s,T\right)\text{d}\tilde{W}\left(s\right)}.\text{ (15.18)}$

By comparing SDEs (15.10) and (15.17), we notice that the bond Z is riskier than the bank account B since the SDE in (15.17) contains an extra Brownian term.

To find the SDE for Z(t, T) under the original (physical) ℙ-measure, we use the equivalence of the measures P and $\tilde{\mathbb{P}}$. By Girsanov’s Theorem 11.13, there exists a change of measure generated by an F-adapted process γ(t) such that

$\tilde{W}\left(t\right)=W\left(t\right)+{\displaystyle {\int }_0^t\gamma \left(s\right)}\text{d}s\text{ (15.19)}$

is a standard $\tilde{\mathbb{P}}$-BM, given W(t) is a standard ℙ-BM, for all t ∊ [0, T*]. Substituting dW(t) + γ(t)dt in place of $\text{d}\tilde{W}\left(t\right)$ in (15.17) gives the following SDE under ℙ:

$\frac{\text{d}Z\left(t,T\right)}{Z\left(t,T\right)}=\left(r\left(t\right)+\gamma \left(t\right){\sigma }_Z\left(t,T\right)\right)\text{d}t+{\sigma }_Z\left(t,T\right)\text{d}W\left(t\right),\text{ (15.20)}$

for all t ∊ [0, T], T ≤ T*. The quantity γ(t) is the excess rate of return over the risk-free rate of return r(t) per one unit of volatility; it is known as the market price of risk or risk premium. This term represents the extra reward we receive for investing in the risky bond rather than in the risk-free bank account. Since γ(t) is adapted to the σ-algebra ℱt generated by the short rate process {r(t)}t≥0 and since {r(t)}t≥0 is a Markov process, the risk premium is a function of t and r(t), i.e., γ = γ(t, r(t)). The market price of risk can be estimated from observable values of rate and bond prices. However, it is a common practice to assume that γ is constant.

15.2.1 Diffusion Models for the Short Rate Process

Suppose the short rate process {r(t)}t≥0 is a diffusion described by an SDE

$\text{d}r\left(t\right)=a\left(t,r\left(t\right)\right)\text{d}t+b\left(t,r\left(t\right)\right)\text{d}W\left(t\right).\text{ (15.21)}$

The coefficients a and b are smooth functions that meet standard conditions required to ensure the existence of solutions to (15.21). Here is a list of popular models of interest rates falling in the class of diffusions as in (15.21):

$\text{d}r\left(t\right)=\alpha \text{d}t+\sigma \text{d}W\left(t\right)\left(\text{the}Mertonmodel\right)\text{ (15.22)}$

$\text{d}r\left(t\right)=\beta r\left(t\right)\text{d}t+\sigma r\left(t\right)\text{d}W\left(t\right)\left(\text{the}Dothanmodel\right)\text{ (15.23)}$

$\text{d}r\left(t\right)=\left(\alpha -\beta r\left(t\right)\right)\text{d}t+\sigma \text{d}W\left(t\right)\left(\text{the}Vasi\text{č}ekmodel\right)\text{ (15.24)}$

$\text{d}r\left(t\right)=\left(\alpha -\beta r\left(t\right)\right)\text{d}t+\sigma r\left(t\right)\text{d}W\left(t\right)\left(\text{the}Brennan-Schwartzmodel\right)\text{ (15.25)}$

$\text{d}r\left(t\right)=\left(\alpha -\beta r\left(t\right)\right)\text{d}t+\sigma \sqrt{r\left(t\right)}\text{d}W\left(t\right)\left(\text{the}Cox-Ingersoll-Rossmodel\right)\text{ (15.26)}$

$\text{d}r\left(t\right)=\alpha \left(t\right)\text{d}t+\sigma \text{d}W\left(t\right)\left(\text{the}Ho-Leemodel\right)\text{ (15.27)}$

$\text{d}r\left(t\right)=\alpha \left(t\right)r\left(t\right)\text{d}t+\sigma \left(t\right)\text{d}W\left(t\right)\left(\text{the}Black-Derman-Toymodel\right)\text{ (15.28)}$

$\text{d}r\left(t\right)=\left(\alpha \left(t\right)-\beta \left(t\right)r\left(t\right)\right)\text{d}t+\sigma \left(t\right)\text{d}W\left(t\right)\left(\text{the}Hull-Whitemodel\right)\text{ (15.29)}$

$\text{d}r\left(t\right)=r\left(t\right)\left(\alpha \left(t\right)-\beta \left(t\right)\ln r\left(t\right)\right)\text{d}t+\sigma \left(t\right)r\left(t\right)\text{d}W\left(t\right)\left(\text{the}Black-Karasinskimodel\right)\text{ (15.30)}$

Here, α, β, and σ are constants; α(t), β(t), and σ(t) are nonrandom continuous functions of time. Assuming that the coefficients a and b in (15.21) are independent of t, i.e., a = a(r(t)) and b = b(r(t)), we obtain a time-homogeneous model for short rates. The models (15.22)–(15.26) are time homogeneous, whereas the models (15.27)–(15.30) are time-inhomogeneous.

Each model has some motivation behind it. There are three main characteristics to be taken into account when a model for interest rates is designed.

• Positiveness of interest rates. For instance, the Merton model and the Vasičcek model do not guarantee that r(t) remains positive. However, these models can still be used for short intervals of time.
• The rate process r(t) is mean reverting, meaning that r(t) fluctuates near a fixed long-term mean level given by limt→∞ E[r(t)]. So the process cannot wander off toward +∞ (or to zero) since a negative drift (or a positive drift) will eventually pull the path back to the long-term level. For instance, the Vasičcek model has the long-term mean level equal to α/β. The drift rate of the diffusion in (15.24) is positive for r(t) < α/β and is negative for r(t) > α/β.
• The model has to be tractable. Ideally, formulae for bond prices and for prices of some derivatives can be derived in closed form.

Note that any model is only an approximation of reality, and a financial model is worthy of consideration if it gives a good approximation of what is observed on the market. The coefficient functions a and b involve parameters that need to be estimated. For instance, they can be chosen so that model values of bonds and other derivatives are as close to the respective market values as possible.

15.2.2 PDE for the Zero-Coupon Bond Value

A model for the term structure of interest rates and bond pricing can be developed from a model for the short rate process. Let us derive the governing differential equation for the zero-coupon bond price. Our first approach employs the (discounted) Feynman–Kac formula. The second method presented here is based on the no-arbitrage argument.

The modelling equation (15.21) for r(t) is specified under the real-world measure. The price Z(t, T) of a zero-coupon bond satisfies (15.11), where the expectation is taken under the EMM $\tilde{\mathbb{P}}$. So we need to know the risk-neutral dynamics of the short rate process. Using (15.19), we change measure in (15.21) to obtain the SDE

$\text{d}r\left(t\right)=\left(a\left(t,r\left(t\right)\right)-\gamma \left(t,r\left(t\right)\right)b\left(t,r\left(t\right)\right)\right)\text{d}t+b\left(t,r\left(t\right)\right)\text{d}\tilde{W}\left(t\right).\text{ (15.31)}$

The short rate process {r(t)}t≥0 is hence Markovian. The expectation conditional on ℱt in the right-hand side of (15.11) is then simply a conditional expectation given the value of r(t). That is, as a random variable the zero-coupon bond price is given as a function of the random variable r(t), Z(t, T) ≡ Z(t, T, r(t)), where

$Z\left(t,T,r\left(t\right)\right)=\tilde{\text{E}}\left[{\text{e}}^{-{\displaystyle {\int }_t^{T}r\left(s\right)}\text{d}s}|{ℱ}_t\right]=\tilde{\text{E}}\left[{\text{e}}^{-{\displaystyle {\int }_t^{T}r\left(s\right)}\text{d}s}|r\left(t\right)\right].$

Conditioning on the known (spot) value of the short rate, r(t) = r, gives the pricing function Z(t, T, r) for a zero-coupon bond. Thus, the time-t value of a zero-coupon bond is an ordinary function (i.e., f(t, r) = Z(t, T, r)) of the current short rate r and time t (for fixed T) where

$Z\left(t,T,r\right)={\tilde{\text{E}}}_{t,r}\left[{\text{e}}^{-{\displaystyle {\int }_t^{T}r\left(s\right)}\text{d}s}\right]\equiv \tilde{\text{E}}\left[{\text{e}}^{-{\displaystyle {\int }_t^{T}r\left(s\right)}\text{d}s}|r\left(t\right)=r\right].\text{ (15.32)}$

By associating this expectation with that in (11.62), the (discounted) Feynman–Kac Theorem 11.9 shows that the pricing function Z = Z(t, T, r) satisfies the following PDE2 (see (11.63)):

$\frac{\partial Z}{\partial t}+\frac{b^2\left(t,r\right)}{2}\frac{{\partial }^{2}Z}{\partial r^2}+\left(a\left(t,r\right)-\gamma \left(t,r\right)b\left(t,r\right)\right)\frac{\partial Z}{\partial r}-rZ=0\text{ (15.33)}$

for all t < T and all r, subject to the terminal condition

$Z\left(T,T,r\right)=1\text{for}\text{all}r.\text{ (15.34)}$

Once the short rate model and the market price of risk γ are specified, the bond price can be determined by solving (15.33) subject to (15.34).

An alternative derivation of the PDE (15.33) is based on hedging and no-arbitrage arguments. Applying the Itô formula to the bond price process Z(t, T) ≡ Z(t, T, r(t)), which is a function of the stochastic short rate r(t) and time t, gives the following SDE for the bond price process:

$\text{d}Z\left(t,T\right)=\left(\frac{\partial Z}{\partial t}+a\frac{\partial Z}{\partial r}+\frac{b^2}{2}\frac{{\partial }^{2}Z}{\partial r^2}\right)\text{d}t+b\frac{\partial Z}{\partial r}\text{d}W\left(t\right)$

or, equivalently, in log-normal form

$\frac{\text{d}Z\left(t,T\right)}{Z\left(t,T\right)}=\mu \text{d}t+\sigma \text{d}W\left(t\right),$

where the log-drift rate µ and log-diffusion coefficient σ are, respectively,

$\mu \left(t,T,r\right)=\frac{1}{Z\left(t,T,r\right)}\left[\frac{\partial Z\left(t,T,r\right)}{\partial t}+a\left(t,r\right)\frac{\partial Z\left(t,T,r\right)}{\partial r}+\frac{b^2\left(t,r\right)}{2}\frac{{\partial }^{2}Z\left(t,T,r\right)}{\partial r^2}\right],\text{ (15.35)}$

$\sigma \left(t,T,r\right)=\frac{b\left(t,r\right)}{Z\left(t,T,r\right)}\frac{\partial Z\left(t,T,r\right)}{\partial }.\text{ (15.36)}$

The underlying short rate process is not a traded security and it cannot therefore be used for hedging bonds. Instead, we try to hedge one bond with another one of a different maturity. Consider two zero-coupon bonds maturing at times T1 and T2 with T1 < T2, respectively. At time t, we buy T1-bonds of value V1(t) (a long position) and sell T2-bonds of value V2(t) (a short position). The total value of this portfolio at time t is Π(t) = V1(t) − V2(t). The change in portfolio value from t to t + dt is

$\begin{array}{c}\text{d}\Pi \left(t\right)=\frac{V_1\left(t\right)}{Z\left(t,T_1\right)}\text{d}Z\left(t,T_1\right)-\frac{V_2\left(t\right)}{Z\left(t,T_2\right)}\text{d}Z\left(t,T_2\right)\\ =V_1\left(t\right)\left({\mu }_1\text{d}t+{\sigma }_1\text{d}W\left(t\right)\right)-V_2\left(t\right)\left({\mu }_2\text{d}t+{\sigma }_2\text{d}W\left(t\right)\right)\\ =\left(V_1\left(t\right){\mu }_1-V_2\left(t\right){\mu }_2\right)\text{d}t+\left(V_1\left(t\right){\sigma }_1-V_2\left(t\right){\sigma }_2\right)\text{d}W\left(t\right),\end{array}$

where, to compact notations, we denote

${\mu }_i=\mu \left(t,T_i,r\left(t\right)\right)\text{and}{\sigma }_i=\sigma \left(t,T_i,r\left(t\right)\right)\text{for}i=1,2.$

Suppose that V1 and V2 are chosen such that

$\frac{V_1\left(t\right)}{V_2\left(t\right)}=\frac{{\sigma }^2}{{\sigma }_1}\equiv \frac{\sigma \left(t,T_2,r\left(t\right)\right)}{\sigma \left(t,T_1,r\left(t\right)\right)}$

holds for all t. Then V1σ1 −V2σ2 ≡ 0 and hence the dW(t) term in the stochastic differential dΠ vanishes. As a result, the equation for dΠ becomes

$\frac{\text{d}\Pi \left(t\right)}{\Pi \left(t\right)}=\frac{{\mu }_1{\sigma }_2-{\mu }_2{\sigma }_1}{{\sigma }_2-{\sigma }_1}\equiv \frac{\mu \left(t,T_1,r\left(t\right)\right)\sigma \left(t,T_2,r\left(t\right)\right)-\mu \left(t,T_2,r\left(t\right)\right)\sigma \left(t,T_1,r\left(t\right)\right)}{\sigma \left(t,T_2,r\left(t\right)\right)-\sigma \left(t,T_1,r\left(t\right)\right)}\text{d}t$

Thus, we have a risk-free, self-financing portfolio strategy. To avoid arbitrage, the rate of return has to be equal to the risk-free rate r(t); that is,

$\frac{{\mu }_1{\sigma }_2-{\mu }_2{\sigma }_1}{{\sigma }_2-{\sigma }_1}=r\iff \frac{{\mu }_1-r}{{\sigma }_1}=\frac{{\mu }_2-r}{{\sigma }_2}.$

The above relation is valid for arbitrary maturity dates T1 and T2, so the ratio $\frac{\mu \left(t,T,r\right)-r}{\sigma \left(t,T,r\right)}$ is independent of T for all T > t. We can hence define

$\gamma \left(t,r\right)=\frac{\mu \left(t,T,r\right)-r}{\sigma \left(t,T,r\right)}\text{ (15.37)}$

so that the drift rate of the bond price process is

$\mu \left(t,T,r\right)=r+\gamma \left(t,r\right)\sigma \left(t,T,r\right).\text{ (15.38)}$

Comparing the above equation with (15.20), we conclude that γ is nothing but the market price of risk for the short rate process. Equating the formulae (15.35) and (15.38) gives the governing PDE (15.33) for the price of a zero-coupon bond.

15.2.3 Affine Term Structure Models

A short-rate model that produces the bond pricing function of the form

$Z\left(t,T,r\right)={\text{e}}^{A\left(t,T\right)-C\left(t,T\right)r},\text{ (15.39)}$

where r is the short rate at time t, and the functions A(t, T) and C(t, T) are independent of r, is called an affine term structure model. Let the short rate process follow the SDE of the form in (15.31):

$\text{d}r\left(t\right)=\tilde{a}\left(t,r\left(t\right)\right)\text{d}t+b\left(t,r\left(t\right)\right)\text{d}\tilde{W}\left(t\right),\text{ (15.40)}$

where $\tilde{a}\left(t,r\right):=a\left(t,r\right)-\gamma \left(t,r\right)b\left(t,r\right)$ and $\tilde{W}\left(t\right)$ is a standard BM under the EMM $\tilde{\mathbb{P}}$.

Certain conditions must be set on the short rate process r(t) in order that the zero-coupon bond price admits the form (15.39). Applying the Itô formula to the bond price Z(t, T, r(t)) = eA(t, T)−C(t, T)r(t), where r(t) follows (15.40), gives

$\begin{array}{c}\frac{\text{d}Z\left(t,T,r\left(t\right)\right)}{Z\left(t,T,r\left(t\right)\right)}=\left[\frac{\partial A\left(t,T\right)}{\partial t}-\frac{\partial C\left(t,T\right)}{\partial t}r\left(t\right)-C\left(t,T\right)\tilde{a}\left(t,r\left(t\right)\right)+\frac{1}{2}{C}^2\left(t,T\right)b^2\left(t,r\left(t\right)\right)\right]\text{d}t\\ -C\left(t,T\right)b\left(t,r\left(t\right)\right)\text{d}\tilde{W}\left(t\right).\end{array}$

We also know that the risk-neutral dynamics of the bond price is given by (15.17). That is, the drift rate in the above SDE has to be equal to the risk-neutral rate r(t). It follows that the ordinary (nonrandom) function g(t, r) defined by

$g\left(t,r\right)=\frac{\partial A\left(t,T\right)}{\partial t}-\frac{\partial C\left(t,T\right)}{\partial t}r-C\left(t,T\right)\tilde{a}\left(t,r\right)+\frac{1}{2}{C}^2\left(t,T\right)b^2\left(t,r\right)-r$

is identically zero for all t and r. Since A and C are independent of r, then g(t, r) ≡ 0 holds only if $\tilde{a}\left(t,r\right)\text{and}{b}^2\left(t,r\right)$ are both linear functions of r. That is, for the bond pricing formula to be of the affine form (15.39) it is necessary that the risk-neutral drift and the square of the diffusion coefficient both be affine (i.e., linear) functions of r:

$\tilde{a}\left(t,r\right)=a_0\left(t\right)+a_1\left(t\right)r\text{and}{b}^2\left(t,r\right)=b_0\left(t\right)+b_1\left(t\right)r\text{ (15.41)}$

where a0(t), a1(t), b0(t), and b1(t) are only functions of time.

The zero-coupon bond pricing function Z(t, T, r) solves the PDE (15.33) with terminal conditions Z(T, T, r) = 1. Substituting the solution in (15.39) into (15.33) yields

$\frac{\partial A\left(t,T\right)}{\partial t}-\left(1+\frac{\partial C\left(t,T\right)}{\partial t}\right)r+\frac{b^2\left(t,r\right)}{2}{C}^2\left(t,T\right)-\tilde{a}\left(t,r\right)C\left(t,T\right)=0\text{for}t<T,\text{ (15.42)}$

with terminal conditions A(T, T) = 0 and C(T, T) = 0.

Substituting (15.41) into (15.42) gives

$\begin{array}{l}\frac{\partial A\left(t,T\right)}{\partial t}-a_0\left(t\right)C\left(t,T\right)+\frac{b_0\left(t\right)}{2}{C}^2\left(t,T\right)\hfill \\ -\left(\frac{\partial C\left(t,T\right)}{\partial t}+a_1\left(t\right)C\left(t,T\right)-\frac{b_1\left(t\right)}{2}{C}^2\left(t,T\right)+1\right)r=0.\hfill \end{array}$

Since the left-hand side of the above equation is identically zero for all values of the rate r, the functions A and C must solve the following pair of differential equations:

$\frac{\partial C\left(t,T\right)}{\partial t}+a_1\left(t\right)C\left(t,T\right)-\frac{b_1\left(t\right)}{2}{C}^2\left(t,T\right)+1=0,\text{ (15.43)}$

$\frac{\partial A\left(t,T\right)}{\partial t}-a_0\left(t\right)C\left(t,T\right)+\frac{b_0\left(t\right)}{2}{C}^2\left(t,T\right)=0,\text{ (15.44)}$

for t < T, subject to the respective boundary conditions at t = T: A(T, T) = 0 and C(T, T) = 0. The equation in (15.43) is a first order nonlinear ODE and is known as the Ricatti equation. For some special cases of a1 and b1, it is possible to solve equation (15.43) in closed form. Once an analytic solution for C is available, the solution for A is obtained by integrating (15.44) with respect to t. In the next three subsections, we consider three short-rate models that admit the bond pricing formula as in (15.39) where A and C are given in analytically closed form.

15.2.4 The Ho–Lee Model

The short rate in the Ho–Lee model satisfies (15.27). We note that the Merton model is a special case of the Ho–Lee model with constant drift rate α. For pricing bonds, we need the short rate dynamics under the EMM $\tilde{\mathbb{P}}$. Assume that the market price of risk γ is independent of r, then by (15.31) the SDE for r(t) under $\tilde{\mathbb{P}}$ is

$\text{d}r\left(t\right)=\tilde{\alpha }\left(t\right)\text{d}t+\sigma \text{d}\tilde{W}\left(t\right),$

where $\tilde{\alpha }(t)=\alpha (t)-\text{γ}\text{(}t\text{)}\sigma$. The strong solution to this linear SDE is a drifted and scaled Brownian motion:

$r\left(s\right)=r\left(t\right)+{\displaystyle {\int }_t^s\tilde{\alpha }}\left(u\right)\text{d}u+\sigma \left(\tilde{W}\left(s\right)-\tilde{W}\left(t\right)\right)\text{ (15.45)}$

for 0 ≤ t ≤ s. The rate r(s) conditional on r(t) = r is normally distributed with mean $r+{\displaystyle {\int }_t^s\tilde{\alpha }}(u)\text{ d}u$ and variance σ2(s – t). Since $\widehat{W}(s-t)=\tilde{W}(s)-\tilde{W}(t)$ is independent of ℱt for every s > t, the random variable ${\displaystyle {\int }_t^T\widehat{W}(s-t)\text{ d}s~Norm(0,{(T-t)}^3/3})$ is independent of ℱt, i.e., it is independent of r(t). Hence, the conditional expectation simplifies to an unconditional expectation and we obtain the no-arbitrage pricing function:

$\begin{array}{c}Z\left(t,T,r\right)={\tilde{\text{E}}}_{t,r}\left[{\text{e}}^{-{\displaystyle {\int }_t^{T}r\left(s\right)\text{d}s}}\right]=\tilde{\text{E}}\left[{\text{e}}^{-{\displaystyle {\int }_t^T\left(r+{\displaystyle {\int }_t^s\tilde{\alpha }}\left(u\right)\text{d}u+\sigma \widehat{W}\left(s-t\right)\right)\text{d}s}}\right]\\ ={\text{e}}^{-r\left(T-t\right)-\widehat{\alpha }\left(t,T\right){\left(T-t\right)}^2/2}\tilde{\text{E}}\left[{\text{e}}^{-\sigma {\displaystyle {\int }_t^T\widehat{W}\left(s-t\right)\text{d}s}}\right]\\ ={\text{e}}^{-r\left(T-t\right)-\widehat{\alpha }\left(t,T\right){\left(T-t\right)}^2/2+{\sigma }^2{\left(T-t\right)}^3/6},\end{array}\text{ (15.46)}$

where $\widehat{\alpha }(t,T):=\frac{2}{{(T-t)}^2}{\displaystyle {\int }_t^T{\displaystyle {\int }_t^s\widehat{\alpha }}(u)\text{d}u\text{d}s}$. In the case of the Merton model with constant $\widehat{\alpha }$, we have $\widehat{\alpha }(t,T)=\widehat{\alpha }$ and

$Z\left(t,T,r\right)={\text{e}}^{-r\left(T-t\right)-\widehat{\alpha }{\left(T-t\right)}^2/2+{\sigma }^2{\left(T-t\right)}^3/6}.\text{ (15.47)}$

The yield rate is an affine function of r,

$y\left(t,T,r\right)=-\frac{\ln Z\left(t,T,r\right)}{T-t}=r+\widehat{\alpha }\left(t,T\right)\frac{\left(T-t\right)}{2}-{\sigma }^2\frac{{\left(T-t\right)}^2}{6}.$

Since the distribution of the short rate r(t) is normal, it then follows that y(t,T,r(t)) is also normally distributed and the distribution of Z(t,T,r(t)) is log-normal.

The Ho–Lee model has several serious shortcomings:

1. the short rate can become negative (with nonzero probability);
2. Z(t,T,r) → ∞ and y(t,T,r) → – ∞ as T → ∞;
3. the yield rates y(t,T1,r(t)) and y(t,T2,r(t)) for different maturities T1 and T2 are both a linear function of the short rate r(t), and they are hence perfectly correlated.

The Ho–Lee model is an affine model with $a_0(t),a_1(t)\equiv 0,b_0(t)={\sigma }^2$, and $b_1(t)\equiv 0$ in (15.41). Therefore, the bond pricing function in (15.47) can also be found by solving (15.43)–(15.44) for A(t,T) and C(t,T), which take the following simpler form:

$\begin{array}{l}\frac{\partial C\left(t,T\right)}{\partial t}+1=0,\hfill \\ \frac{\partial A\left(t,T\right)}{\partial t}-\tilde{\alpha }\left(t\right)C\left(t,T\right)+\frac{{\sigma }^2}{2}{C}^2\left(t,T\right)=0,\hfill \end{array}$

subject to A(T,T) = C(T,T) = 0. The first equation is trivially integrated to give C(t,T) = T–t. Substituting this into the second equation, integrating and applying the condition A(T,T) = 0, gives

$A\left(t,T\right)=-{\displaystyle {\int }_t^T\tilde{\alpha }}\left(u\right)\left(T-u\right)\text{d}u+\frac{{\sigma }^2}{2}{\displaystyle {\int }_t^T{\left(T-u\right)}^2}\text{d}u=-\widehat{\alpha }\left(t,T\right)\frac{{\left(T-t\right)}^2}{2}+{\sigma }^2\frac{{\left(T-t\right)}^3}{6}.$

This recovers the above same formula for the yield and zero-coupon bond price in (15.47). For the case of constant $\tilde{\alpha }(t)\equiv \tilde{\alpha }$, the formulae simplify where $\tilde{\alpha }(t,T)=\tilde{\alpha }$.

15.2.5 The Vasiček Model

The short rate process in the Vasicek model satisfies SDE (15.24). Recall that the solution to (15.24) is also known as the Ornstein–Uhlenbeck process. Assuming the market price of risk γ is constant, we obtain the following risk-neutral dynamics of r(t):

$\text{d}r\left(t\right)=\left(\tilde{\alpha }-\beta r\left(t\right)\right)\text{d}t+\sigma \text{d}\tilde{W}\left(t\right),\text{ (15.48)}$

where $\tilde{\alpha }=\alpha -\gamma \sigma ,\beta ,\text{and }\sigma$ are positive parameters. The diffusion solving the above SDE is called a mean-reverting process since the instantaneous drift $\tilde{\alpha }-\beta r(t)=\beta (\tilde{\alpha }/\beta -r(t))$ pulls the process toward the constant mean level $\tilde{\alpha }/\beta$ with magnitude proportional to the deviation of the process from the mean.

Integrating the above SDE (see Examples 11.8 and 11.13) gives

$r\left(T\right)={\text{e}}^{-\beta \left(T-t\right)}r\left(t\right)+\frac{\tilde{\alpha }}{\beta }\left(1-{\text{e}}^{-\beta \left(T-t\right)}\right)+\sigma {\displaystyle {\int }_t^T{\text{e}}^{-\beta \left(T-s\right)}\text{d}\tilde{W}\left(s\right)}.\text{ (15.49)}$

The probability distribution of r(T) conditional on r(t) is normal with mean

$\tilde{\text{E}}\left[r\left(T\right)|r\left(t\right)\right]={\text{e}}^{-\beta \left(T-t\right)}r\left(t\right)+\frac{\tilde{\alpha }}{\beta }\left(1-{\text{e}}^{-\beta \left(T-t\right)}\right)$

and variance

$\widetilde{\text{Var}}\left(r\left(T\right)|r\left(t\right)\right)={\sigma }^2\frac{1-{\text{e}}^{-2\beta \left(T-t\right)}}{2\beta }.$

Note that the long-term mean and variance (as T – t → ∞) are given by the constants

$\underset{T\to \infty }{\lim }\tilde{\text{E}}\left[r\left(T\right)|r\left(t\right)\right]=\frac{\tilde{\alpha }}{\beta }\text{and}\underset{T\to \infty }{\lim }\widetilde{\text{Var}}\left(r\left(T\right)|r\left(t\right)\right)=\frac{{\sigma }^2}{2\beta }.$

We now proceed to the calculation of bond prices. The drift and diffusion coefficients in (15.48) correspond to $a_0=\tilde{\alpha },a_1=-\beta ,b_0={\sigma }^2$, and b1 = 0 in (15.41). We have the following pair of ODEs for A(t,T) and C(t,T):

$\frac{\partial C\left(t,T\right)}{\partial t}-\beta C\left(t,T\right)+1=0,\text{ (15.50)}$

$\frac{\partial A\left(t,T\right)}{\partial t}-\tilde{\alpha }C\left(t,T\right)+\frac{{\sigma }^2}{2}{C}^2\left(t,T\right)=0,\text{ (15.51)}$

for t < T, subject to A(T,T) = C(T,T) = 0. Solving the above system, we obtain

$C\left(t,T\right)=\frac{1}{\beta }\left(1-{\text{e}}^{-\beta \left(T-t\right)}\right)\text{and}A\left(t,T\right)=\left(C\left(t,T\right)-\left(T-t\right)\right)y_{\infty }-\frac{{\sigma }^2}{4\beta }{C}^2\left(t,T\right)\text{ (15.52)}$

where $y_{\infty }:=\frac{\tilde{\alpha }}{\beta }-\frac{{\sigma }^2}{2{\beta }^2}$ Thus, according to (15.39), the bond pricing formula for the Vasiček model is

$Z\left(t,T,r\right)=\exp \left[\left(\frac{1-{\text{e}}^{-\beta \left(T-t\right)}}{\beta }\right)\left(y_{\infty }-r\right)-y_{\infty }\left(T-t\right)-\frac{{\sigma }^2}{4\beta }{\left(\frac{1-{\text{e}}^{-\beta \left(T-t\right)}}{\beta }\right)}^2\right].\text{ (15.53)}$

The yield rate is found to be

$y\left(t,T,r\right)=y_{\infty }-\frac{1}{\beta }\left(1-{\text{e}}^{-\beta \left(T-t\right)}\right)\frac{y_{\infty }-r}{T-t}+\frac{{\sigma }^2}{4\beta \left(T-t\right)}{\left(\frac{1-{\text{e}}^{-\beta \left(T-t\right)}}{\beta }\right)}^2.$

By taking T → ∞, the last two terms of the above equation vanish so that the long-term yield rate is in fact equal to y∞.

15.2.6 The Cox–Ingersoll–Ross Model

One common drawback of the Merton model and the Vasiček model is that the short rate can be negative due to its normal distribution. The first tractable model for the short rate process r(t) that keeps rates positive was proposed by Cox, Ingersoll, and Ross (the CIR model). The short-rate model follows the square root diffusion described by the SDE in (15.26). Assuming the market price of risk is equal to $\gamma \sqrt{r}$, with constant γ, we obtain the following risk-neutral dynamics:

$\text{d}r\left(t\right)=\left(\alpha -\tilde{\beta }r\left(t\right)\right)\text{d}t+\sigma \sqrt{r\left(t\right)}\text{d}\tilde{W}\left(t\right).$

where $\tilde{\beta }=\beta +\gamma \sigma$. With a nonnegative initial interest rate, r(t) stays nonnegative. Moreover, the CIR process is mean-reverting with the long-run mean level $\alpha /\tilde{\beta }$.

The CIR process r(t) is reduced to the squared Bessel (SQB) process X(t), which solves the SDE (16.14) by means of a scale and time transformation,

$r\left(t\right)={\text{e}}^{-\tilde{\beta }t}\frac{{\sigma }^2}{4}X\left({\tau }_t\right)$

where the (strictly increasing) time transformation τt is defined to be

${\tau }_t:=\{\begin{array}{l}t\text{if}\tilde{\beta }=0,\hfill \\ \frac{{\text{e}}^{\tilde{\beta }t}-1}{\tilde{\beta }}\text{if}\tilde{\beta }\ne 0.\hfill \end{array}\text{ (15.54)}$

The index of the SQB process in (16.14) is $\mu =\frac{2\alpha }{{\sigma }^2}-1$. In what follows, we assume that $\mu \ge 0$ (or, equivalently, $2\alpha \ge {\sigma }^2$) and hence the left-hand endpoint 0 is an entrance boundary for the short-rate CIR process.

The transition PDF for the CIR process relates to that of the SQB process. Under the risk-neutral measure $\tilde{\mathbb{P}}$ it is given by

$\tilde{p}\left(t;r_0,r\right)=c_t{\text{e}}^{\tilde{\beta }t}{\left(\frac{r{\text{e}}^{\tilde{\beta }t}}{r_0}\right)}^{\frac{\mu }{2}}\exp \left(-c_t\left(r{\text{e}}^{\tilde{\beta }t}+r_0\right)\right)I_{\mu }\left(2c_t\sqrt{r{r}_0{\text{e}}^{\tilde{\beta }t}}\right),\text{ (15.55)}$

where $c_t:=\frac{2}{{\sigma }^2{\tau }_t}$.

By expressing the above SDE in integral form, with r(0) = r0, and taking expectations (under measure $\tilde{\mathbb{P}}$) on both sides while denoting the mean by $\tilde{\text{E}}[r(t)]\equiv m(t)$, we have

$m\left(t\right)=r_0+{\displaystyle {\int }_0^t\left(\alpha -\tilde{\beta }m\left(u\right)\right)\text{d}u.}$

Here we used the fact that the Itô integral ${\displaystyle {\int }_0^t\sqrt{r\left(u\right)}\text{d}\tilde{W}\left(u\right)}$ has zero expectation (i.e., it is a $\tilde{\mathbb{P}}\text{-martingale}$). Differentiating gives a linear first order ODE

$\frac{\text{d}}{\text{d}t}m\left(t\right)=\alpha -\tilde{\beta }m\left(t\right),t\ge 0,$

subject to m(0) = r0. Solving gives

$m\left(t\right)\equiv \tilde{\text{E}}\left[r\left(t\right)\right]=\frac{\alpha }{\tilde{\beta }}\left(1-{\text{e}}^{-\tilde{\beta }t}\right)+r_0{\text{e}}^{-\tilde{\beta }t}.$

Here, we assume that $\tilde{\beta }\ne 0$. Note that the mean of the process converges to the long-term level $\alpha /\tilde{\beta }$, as t → ∞.

The CIR model is within the class of affine term structure models with a bond pricing formula of the form in (15.39). The respective ODEs in (15.43) and (15.44) for A(t,T) and C(t,T) are

$\frac{\partial C\left(t,T\right)}{\partial t}-\tilde{\beta }C\left(t,T\right)-\frac{{\sigma }^2}{2}{C}^2\left(t,T\right)+1=0,\text{ (15.56)}$

$\frac{\partial A\left(t,T\right)}{\partial t}-\alpha C\left(t,T\right)=0,\text{ (15.57])}$

subject to A(T,T) = C(T,T) = 0. The first equation is a first order ODE with a quadratic nonlinear term. The trick in solving this equation is to turn it into a linear second order ODE by invoking the transformation

$\psi \left(t\right):=\exp \left(\frac{{\sigma }^2}{2}{\displaystyle {\int }_t^{T}C\left(s,T\right)\text{d}s}\right).$

Taking derivatives gives (denoting $C^{\prime }(t,T)\equiv \partial C(t,T)/\partial t$:

$\begin{array}{l}\frac{{\psi }^{\prime }\left(t\right)}{\psi \left(t\right)}\equiv \frac{\text{d}}{\text{d}t}\ln \psi \left(t\right)=-\frac{{\sigma }^2}{2}C\left(t,T\right)\Rightarrow C\left(t,T\right)=-\frac{2}{{\sigma }^2}\frac{{\psi }^{\prime }\left(t\right)}{\psi \left(t\right)},\hfill \\ \frac{{\psi }^{″}\left(t\right)}{\psi \left(t\right)}=-\frac{{\sigma }^2}{2}{C}^{\prime }\left(t,T\right)-\frac{{\sigma }^2}{2}C\left(t,T\right)\frac{{\psi }^{\prime }\left(t\right)}{\psi \left(t\right)}\Rightarrow C^{\prime }\left(t,T\right)=-\frac{2}{{\sigma }^2}\frac{{\psi }^{″}\left(t\right)}{\psi \left(t\right)}+\frac{{\sigma }^2}{2}{C}^2\left(t,T\right).\hfill \end{array}$

Substituting the above two expressions for C(t,T) and C' (t,T) into (15.56), and simplifying, gives

${\psi }^{″}\left(t\right)-\tilde{\beta }{\psi }^{\prime }\left(t\right)-\frac{{\sigma }^2}{2}\psi \left(t\right)=0.\text{ (15.58)}$

This is a second order linear ODE with constant coefficients. Its solution is found by standard methods. In particular, this ODE has the general solution

$\psi \left(t\right)=c_1{\text{e}}^{\frac{1}{2}\left(\tilde{\beta }+\vartheta \right)t}+c_2{\text{e}}^{\frac{1}{2}\left(\tilde{\beta }-\vartheta \right)t}$

with derivative

${\psi }^{\prime }\left(t\right)=\frac{c_1}{2}\left(\tilde{\beta }+\vartheta \right){\text{e}}^{\frac{1}{2}\left(\tilde{\beta }+\vartheta \right)t}+\frac{c_2}{2}\left(\tilde{\beta }-\vartheta \right){\text{e}}^{\frac{1}{2}\left(\tilde{\beta }-\vartheta \right)t},$

where $\vartheta :=\sqrt{{\tilde{\beta }}^2+2{\sigma }^2}$. The constants c1,2 are determined by applying the boundary conditions, ${\psi }^{\prime }(T)={\text{e}}^{\text{0}}=1\text{ and }{\psi }^{\prime }(T)=-\frac{{\sigma }^2}{2}C(T,T)\psi (T)=0$, giving a 2 × 2 linear system in c1 and c2:

$\begin{array}{l}{\text{e}}^{\frac{1}{2}\left(\tilde{\beta }+\vartheta \right)T}{c}_1+{\text{e}}^{\frac{1}{2}\left(\tilde{\beta }-\vartheta \right)T}{c}_2=1\hfill \\ \frac{1}{2}\left(\tilde{\beta }+\vartheta \right){\text{e}}^{\frac{1}{2}\left(\tilde{\beta }+\vartheta \right)T}{c}_1+\frac{1}{2}\left(\tilde{\beta }-\vartheta \right){\text{e}}^{\frac{1}{2}\left(\tilde{\beta }-\vartheta \right)T}{c}_2=0.\hfill \end{array}$

Solving gives

$c_1=\left(\frac{1}{2}-\frac{\tilde{\beta }}{2\vartheta }\right){\text{e}}^{-\frac{1}{2}\left(\tilde{\beta }+\vartheta \right)T}\text{and}{c}_2=\left(\frac{1}{2}+\frac{\tilde{\beta }}{2\vartheta }\right){\text{e}}^{-\frac{1}{2}\left(\tilde{\beta }-\vartheta \right)T}.$

Using these coefficients gives the unique explicit expression for $\psi (t)$, which can be equiva-lently written as

$\begin{array}{l}\psi \left(t\right)=\left(\frac{1}{2}-\frac{\tilde{\beta }}{2\vartheta }\right){\text{e}}^{-\frac{1}{2}\left(\tilde{\beta }+\vartheta \right)\left(T-t\right)}+\left(\frac{1}{2}+\frac{\tilde{\beta }}{2\vartheta }\right){\text{e}}^{-\frac{1}{2}\left(\tilde{\beta }-\vartheta \right)\left(T-t\right)}\hfill \\ ={\text{e}}^{-\frac{1}{2}\tilde{\beta }\left(T-t\right)}\left[\cosh \frac{\vartheta \left(T-t\right)}{2}+\frac{\tilde{\beta }}{\vartheta }\sinh \frac{\vartheta \left(T-t\right)}{2}\right].\hfill \end{array}$

Differentiating this expression, and using the fact that $C\left(t,T\right)=-\frac{2}{{\sigma }^2}\frac{\psi \prime \left(t\right)}{\psi \left(t\right)},$ finally gives

$C\left(t,T\right)=\frac{2\sinh \frac{\vartheta \left(T-t\right)}{2}}{\vartheta \cosh \frac{\vartheta \left(T-t\right)}{2}+\tilde{\beta }\sinh \frac{\vartheta \left(T-t\right)}{2}}=\frac{2{\text{e}}^{\vartheta \left(T-t\right)}-2}{\left(\tilde{\beta }+\vartheta \right)\left({\text{e}}^{\vartheta \left(T-t\right)}-1\right)+2\vartheta }.\text{ (15.59)}$

Having solved for C(t, T), the function A(t, T) is obtained by integrating (15.57), with ${\displaystyle {\int }_t^T\frac{\partial A\left(u,T\right)}{\partial u}\text{d}u=A\left(T,T\right)-A\left(t,T\right)=-A\left(t,T\right),}$ giving

$A\left(t,T\right)=-\alpha {\displaystyle {\int }_t^{T}C\left(u,T\right)}\text{d}u.$

We can compute this integral using the fact that $-C\left(t,T\right)=\frac{2}{{\sigma }^2}\frac{\text{d}}{\text{d}t}\ln \psi \left(t\right),$ i.e.,

$A\left(t,T\right)=\frac{2\alpha }{{\sigma }^2}{\displaystyle {\int }_t^T\frac{\text{d}}{\text{d}u}}\ln \psi \left(u\right)\text{d}u=\frac{2\alpha }{{\sigma }^2}\ln \frac{\psi \left(T\right)}{\psi \left(t\right)}=\frac{2\alpha }{{\sigma }^2}\ln \frac{1}{\psi \left(t\right)}.$

Note that ψ(T) = 1. Using the above expression for ψ gives the explicit form for A(t, T), which we can write equivalently as

$A\left(t,T\right)=\frac{2\alpha }{{\sigma }^2}\ln \left(\frac{\vartheta {\text{e}}^{\frac{1}{2}\tilde{\beta }\left(T-t\right)}}{\vartheta \cosh \frac{\vartheta \left(T-t\right)}{2}+\tilde{\beta }\sinh \frac{\vartheta \left(T-t\right)}{2}}\right)=\frac{2\alpha }{{\sigma }^2}\ln \left(\frac{2\vartheta {\text{e}}^{\frac{1}{2}\left(\tilde{\beta }+\vartheta \right)\left(T-t\right)}}{\left(\tilde{\beta }+\vartheta \right)\left({\text{e}}^{\vartheta \left(T-t\right)}-1\right)+2\vartheta }\right).\text{ (15.60)}$

Inserting the coefficients in (15.60) and (15.59) into (15.39) gives us the closed form analytical expression for the price of a zero-coupon bond under the CIR model.

15.3 Heath–Jarrow–Morton Formulation

When we use a short-rate model such as one of those considered in previous subsections, the bond prices, yield rates, and forward rates are the output of the model. We usually find that the model bond prices Z(t, T) do not perfectly match the market (observed) bond prices Zobs(t, T). The usual approach is to calibrate the short-rate model parameters to achieve the best possible agreement between the model prices and the market prices. For example, we can use the least squares method and find the optimal model parameters by minimizing Σi[Z(t, Ti) − Zobs(t, Ti]2, the sum of squares of the differences between the model and observed bond prices across a given set of maturities T1, T2,.... However, since the number of bonds with different maturities exceeds the number of model parameters, we will find that the observed prices are closer to the prices produced by the calibrated model but still differ to some extent. This issue can be dealt with by using one of the following two approaches. One way is to construct a time-inhomogeneous model with multiple random factors. Such a model will be more flexible than a single-factor model with constant parameters and can be calibrated to historical data with a greater degree of accuracy. Multifactor models are briefly discussed further in Section 15.4. Several time-inhomogeneous short-rate models, including the Ho–Lee model, the Black–Derman-Toy model, and the Hull–White model, are presented in the beginning of the previous section.

The other approach is to construct a no-arbitrage term-structure model where observed bond prices, yield rates, or forward rates are taken as input variables. The ideal result would be if the model prices Z(t, T) precisely match the market prices Zobs(t, T) at the time of calibration t. The Heath–Jarrow–Morton (HJM) framework attempts to construct a model for the family of forward rate curves {f(t, T)}0≤t≤T with 0 ≤ T ≤ T*. Under the HJM model, the forward rate f(t, T) (for arbitrarily fixed T ∊ [0, T*]) follows the SDE

$\text{d}f\left(t,T\right)={\alpha }_F\left(t,T\right)\text{d}t+{\sigma }_F\left(t,T\right)\text{d}W\left(t\right),0\le t\le T,\text{ (15.61)}$

where {W(t)}t≥0 is a standard Brownian motion under the physical measure ℙ, and the processes αF(t, T) and σF(t, T) are adapted to its natural filtration $\mathbb{F}$W. We note that the differential is taken w.r.t. the calendar time variable t and T acts as a fixed parameter. To simplify the analysis in what follows, we shall assume a single Brownian motion where the forward rate process starts with the initial rate f(0, T). The framework readily extends to the case where the forward rates are driven by a multidimensional Brownian motion.

Let us assume that there exists an EMM (risk-neutral measure) $\tilde{\mathbb{P}}$ such that the discounted bond price process ${\left\{\overline{Z}\left(t,T\right)\right\}}_{0\le t\le T}$ is a $\tilde{\mathbb{P}}$-martingale. As is demonstrated below, the EMM assumption implies that the drift coefficient αF is determined by the diffusion coefficient σF when the SDE (15.61) for forward rates is considered under $\tilde{\mathbb{P}}$.

15.3.1 HJM under Risk-Neutral Measure

Let us work out the risk-neutral dynamics of the bond prices. The zero-coupon bond price can be expressed in terms of forward rates, as given in (15.7). The discounted bond price is

$\overline{Z}\left(t,T\right)\equiv D\left(t\right)Z\left(t,T\right)=\exp \left(-{\displaystyle {\int }_0^{t}r\left(u\right)\text{d}u-{\displaystyle {\int }_t^{T}f\left(t,u\right)}\text{d}u}\right),$

for 0 ≤ t ≤ T ≤ T*. By the Itô formula, we have

$\text{d}\overline{Z}\left(t,T\right)=\overline{Z}\left(t,T\right)\left(\text{d}X\left(t\right)+\frac{1}{2}\text{d}\left[X,X\right]\left(t\right)-r\left(t\right)\text{d}t\right),\text{ (15.62)}$

where X(t) denotes the log-price of the bond and is given by

$X\left(t\right):=\ln Z\left(t,T\right)=-{\displaystyle {\int }_t^{T}f\left(t,u\right)\text{d}u.}\text{ (15.63)}$

In (15.62) we used the Itô formula ${\text{de}}^{Y\left(t\right)}={\text{e}}^{Y\left(t\right)}\left(\text{d}Y\left(t\right)+\frac{1}{2}\text{d}\left[Y,Y\right]\left(t\right)\right),\text{where}Y\left(t\right)=X\left(t\right)-{\displaystyle {\int }_0^{t}r\left(u\right)\text{d}u,\text{d}Y\left(t\right)=\text{d}X\left(t\right)-r\left(t\right)\text{d}t,}$ where $Y\left(t\right)=X\left(t\right)-{\displaystyle {\int }_0^{t}r\left(u\right)\text{d}u,\text{d}Y\left(t\right)=\text{d}X\left(t\right)-r\left(t\right)\text{d}t,\text{and}}\text{d}\left[Y,Y\right]\left(t\right)=\text{d}\left[X,X\right]\left(t\right)=\text{d}X\left(t\right)\text{d}X\left(t\right).$

The next step is to derive an SDE for the discounted bond price in terms of the drift and diffusion functions that are driving the forward rate in (15.61). We need to compute the stochastic differential of X(t) defined by (15.63), which is a Riemann integral of the process f(t, u) w.r.t. the parameter u ∊ [t, T]. It can be shown that3

$\text{d}X\left(t\right)\equiv \text{d}\left(-{\displaystyle {\int }_t^{T}f\left(t,u\right)\text{d}}u\right)=f\left(t,t\right)\text{d}t-{\displaystyle {\int }_t^T\text{d}f\left(t,u\right)\text{d}}u.\text{ (15.64)}$

The first term is r(t) dt since the forward rate corresponds to the instantaneous short rate at time t when T = t, i.e., f(t,t) = r(t) is the observed rate on any risk-free investment at current time t. The second term can now be written as a stochastic differential of an Itô process upon substituting the form in (15.61), using the parameter u in the place of T, within the integral and interchanging the order of the differentials (as follows by applying Fubini’s Theorem twice):

$\begin{array}{c}{\displaystyle {\int }_t^T\text{d}f\left(t,u\right)\text{d}}u={\displaystyle {\int }_t^T\left({\alpha }_F\left(t,u\right)\text{d}t+{\sigma }_F\left(t,u\right)\text{d}W\left(t\right)\right)\text{d}u}\\ =\left({\displaystyle {\int }_t^T{\alpha }_F\left(t,u\right)\text{d}}u\right)\text{d}t+\left({\displaystyle {\int }_t^T{\alpha }_F\left(t,u\right)\text{d}}u\right)\text{d}W\left(t\right).\end{array}$

Defining the new drift and volatility functions (which are proportional the instantaneous drift and volatility functions of the forward price process averaged over all maturity times between t and T),

$A_F\left(t,T\right):={\displaystyle {\int }_t^T{\alpha }_F}\left(t,u\right)\text{d}u\text{and}{\displaystyle {\sum }_F\left(t,T\right)}:={\displaystyle {\int }_t^T{\alpha }_F}\left(t,u\right)\text{d}u,\text{ (15.65)}$

gives, according to (15.64),

$\text{d}X\left(t\right)=\left[r\left(t\right)-A_F\left(t,T\right)\right]\text{d}t-{\displaystyle {\sum }_F\left(t,T\right)}\text{d}W\left(t\right).\text{ (15.66)}$

We note that, by differentiating the integrals in (15.65) w.r.t. the maturity variable T, we have

$\frac{\partial }{\partial T}{A}_F\left(t,T\right)={\alpha }_F\left(t,T\right)\text{and}\frac{\partial }{\partial T}{\displaystyle {\sum }_F\left(t,T\right)}={\sigma }_F\left(t,T\right).\text{ (15.67)}$

Therefore, using (15.66) gives d[X, X](t) = ${\Sigma }_F^2\left(t,T\right)\text{d}t$ and the SDE in (15.62) for the discounted bond price takes the form

$\frac{\text{d}\overline{Z}\left(t,T\right)}{\overline{Z}\left(t,T\right)}=\left(\frac{1}{2}{\displaystyle {\sum }_F^2\left(t,T\right)}-A_F\left(t,T\right)\right)\text{d}t-{\displaystyle {\sum }_F\left(t,T\right)}\text{d}W\left(t\right).\text{ (15.68)}$

Note that this SDE is in terms of the Brownian increment in the physical measure ℙ.

The HJM model includes a zero-coupon bond for each maturity T ∊ [0, T*]. Hence, to avoid arbitrage when trading in any of these bonds we make recourse to the First Fundamental Theorem of Chapter 13. Namely, if there exists of a risk-neutral measure $\tilde{\mathbb{P}}$ under which all discounted bond price processes are $\tilde{\mathbb{P}}$-martingales, then there are no arbitrage strategies in the model. The risk-neutral measure $\tilde{\mathbb{P}}$ is the measure under which ${\left\{\overline{Z}\left(t,T\right)\right\}}_{0\le t\le T}$ are martingales for all choices of T ∊ [0, T*]. By Girsanov’s Theorem 11.13, we can change measures from ℙ to $\tilde{\mathbb{P}}$ where

$\tilde{W}\left(t\right):=W\left(t\right)+{\displaystyle {\int }_0^t\theta \left(s\right)}\text{d}s,t\ge 0,$

is a standard $\tilde{\mathbb{P}}$-BM with ${\left\{\theta \left(t\right)\right\}}_{0\le t\le T}$ as an adapted process. Using the differential form $dW\left(t\right)=d\tilde{W}\left(t\right)-\theta \left(t\right)\text{d}t$ into (15.68) gives

$\frac{\text{d}\overline{Z}\left(t,T\right)}{\overline{Z}\left(t,T\right)}=\left(\frac{1}{2}{\displaystyle {\sum }_F^2\left(t,T\right)-A_F\left(t,T\right)+{\displaystyle {\sum }_F\left(t,T\right)}\theta \left(t\right)}\right)\text{d}t-{\displaystyle {\sum }_F\left(t,T\right)}\text{d}\tilde{W}\left(t\right).\text{ (15.69)}$

Hence, the measure $\tilde{\mathbb{P}}$ defines the risk-neutral measure if the drift term in this expression is identically zero, i.e., if θ(t) satisfies the equation

$\frac{1}{2}{\displaystyle {\sum }_F^2\left(t,T\right)}-A_F\left(t,T\right)+{\displaystyle {\sum }_F\left(t,T\right)}\theta \left(t\right)=0,\text{ (15.70)}$

for all 0 ≤ t ≤ T, and for each T ∊ [0, T*]. As we saw in Chapter 11, this corresponds to a so-called market price of risk equation where the unknown θ(t) is the market price of risk. Here, we note that θ(t) must solve the above equation for each maturity value T ∊ [0, T*]. By taking partial derivatives w.r.t. the maturity T on both sides of (15.70), and using the derivatives defined in (15.67), gives:

${\alpha }_F\left(t,T\right)={\sigma }_F\left(t,T\right)\left[{\displaystyle {\sum }_F\left(t,T\right)+\theta \left(t\right)}\right].\text{ (15.71)}$

In order to have no arbitrage in the HJM model, the drift and volatility functions and the averaged volatility ΣF(t, T) for the forward price process must necessarily satisfy a relation of this form for every maturity value T. In fact, if the relation in (15.71) holds, then the HJM model has no arbitrage, and assuming a nonzero volatility function σF(t, T), the risk-neutral measure is given uniquely by solving for θ(t) in (15.71):

$\theta \left(t\right)=\frac{{\alpha }_F\left(t,T\right)}{{\sigma }_F\left(t,T\right)}-{\displaystyle {\sum }_F\left(t,T\right)}\text{ (15.72)}$

for all t ∊ [0, T]. This is the statement of the so-called Heath-Jarrow-Morton Theorem, which states that the above HJM model driven by a single Brownian motion admits no arbitrage if there exists an adapted process θ(t) solving (15.71) for all values of time t and maturities T.

The Heath-Jarrow-Morton Theorem is now readily proven by showing that (15.71), for all 0 ≤ t ≤ T ≤ T*, implies (15.70), i.e., that $\tilde{\mathbb{P}}$ exists. Moreover, if σF (t, T) ≠ 0, then the risk-neutral measure $\tilde{\mathbb{P}}$ is uniquely specified by (15.72). Indeed, by assumption, (15.71) holds if we replace the maturity T by the (dummy) variable s, 0 ≤ s ≤ T ≤ T*, i.e.,

${\alpha }_F\left(t,s\right)={\sigma }_F\left(t,s\right)\left[{\displaystyle \sum \left(t,s\right)+\theta \left(t\right)}\right].$

Integrating both sides of this equation w.r.t. s, from s = t to s = T > t, while fixing t:

$\begin{array}{c}{\displaystyle {\int }_t^T{\alpha }_F\left(t,s\right)\text{d}s}={\displaystyle {\int }_t^T{\sigma }_F\left(t,s\right){\displaystyle {\sum }_F\left(t,s\right)}\text{d}s+\theta \left(t\right)}{\displaystyle {\int }_t^T{\sigma }_F\left(t,s\right)\text{d}s}\\ ={\displaystyle {\int }_t^T\frac{1}{2}\frac{\partial }{\partial s}\left({\displaystyle {\sum }_F^2\left(t,s\right)}\right)\text{d}s+\theta \left(t\right)}{\displaystyle {\int }_t^T\frac{\partial }{\partial s}{\displaystyle {\sum }_F\left(t,T\right)}\text{d}s}\\ =\frac{1}{2}\left[{\displaystyle {\sum }_F^2\left(t,T\right)-}{\displaystyle {\sum }_F^2\left(t,t\right)}\right]+\theta \left(t\right)\left[{\displaystyle {\sum }_F\left(t,T\right)}-{\displaystyle {\sum }_F\left(t,t\right)}\right]\\ =\frac{1}{2}{\displaystyle {\sum }_F^2\left(t,T\right)+\theta \left(t\right)}{\displaystyle {\sum }_F\left(t,T\right)}\end{array}$

where ΣF(t, t) ≡ 0. The integral on the left-hand side is, by definition, AF(t, T), and hence we recover the relation in (15.70).

Hence, assuming no arbitrage in the HJM model, i.e., assuming (15.71) holds, the SDE in (15.69) is driftless, i.e.,

$\frac{\text{d}\overline{Z}\left(t,T\right)}{\overline{Z}\left(t,T\right)}=-{\displaystyle {\sum }_F\left(t,T\right)}\text{d}\tilde{W}\left(t\right),\text{ (15.73)}$

where discounted zero-coupon bond price processes of all maturities T are $\tilde{\mathbb{P}}$-martingales. In particular, the ratio $\frac{\overline{Z}\left(t,T\right)}{\overline{Z}\left(0,T\right)}$ is the stochastic exponential of the process {−ΣF(t, T)}0≤t≤T w.r.t. the Brownian motion $\tilde{W}$ on the time interval [0, t]:

$\begin{array}{l}\overline{Z}\left(t,T\right)=\overline{Z}\left(0,T\right){\epsilon }_t\left(-{\displaystyle {\sum }_F\cdot \tilde{W}}\right)\hfill \\ =Z\left(0,T\right)\exp \left[-\frac{1}{2}{\displaystyle {\int }_0^t{\displaystyle {\sum }_F^2\left(s,T\right)}\text{d}s-{\displaystyle {\int }_0^t{\displaystyle {\sum }_F\left(s,T\right)\text{d}\tilde{W}\left(s\right)}}}\right].\hfill \end{array}\text{ (15.74)}$

Note that $\overline{Z}\left(0,T\right)=Z\left(0,T\right)$. We recall that the unit bank account has value $B\left(t\right)=\text{exp}\left({\displaystyle {\int }_0^{t}r\left(s\right)\text{d}s}\right).$ Hence, the risk-neutral value process o a zero-coupon bond takes the form

$Z\left(t,T\right)=Z\left(0,T\right)\exp \left[-\frac{1}{2}{\displaystyle {\int }_0^t{\displaystyle {\sum }_F^2\left(s,T\right)}\text{d}s-{\displaystyle {\int }_0^t{\displaystyle {\sum }_F\left(s,T\right)}\text{d}\tilde{W}\left(s\right)}+{\displaystyle {\int }_0^{t}r\left(s\right)}\text{d}s}\right].\text{ (15.75)}$

The SDE for Z(t, T) under $\tilde{\mathbb{P}}$ has the form

$\frac{\text{d}Z\left(t,T\right)}{Z\left(t,T\right)}=r\left(t\right)\text{d}t-{\displaystyle {\sum }_F\left(t,T\right)\text{d}\tilde{W}\left(t\right)}.\text{ (15.76)}$

We recall from Chapter 13 that a (domestic) nondividend paying asset has (log-)drift equal to r(t) under the risk-neutral measure. Clearly, a zero-coupon bond is an example of a nondividend paying asset.

Finally, upon using (15.71), the SDE in (15.61) for the forward price process takes the form (w.r.t. the $\tilde{\mathbb{P}}$-BM):

$\begin{array}{c}\text{d}f\left(t,T\right)=\left[{\alpha }_F\left(t,T\right)-\theta \left(t\right){\sigma }_F\left(t,T\right)\right]\text{d}t+{\sigma }_F\left(t,T\right)\text{d}\tilde{W}\left(t\right)\\ ={\sigma }_F\left(t,T\right){\displaystyle {\sum }_F\left(t,T\right)}\text{d}t+{\sigma }_F\left(t,T\right)\text{d}\tilde{W}\left(t\right).\end{array}\text{ (15.77)}$

This SDE shows us that the instantaneous (time-t) risk-neutral drift of the forward price process (for given maturity T) is determined by its instantaneous volatility and its (integrated) volatility across all times up to the maturity T. That is, the forward price has SDE of the form

$\text{d}f\left(t,T\right)={\tilde{\alpha }}_F\left(t,T\right)\text{d}t+{\sigma }_F\left(t,T\right)\text{d}\tilde{W}\left(t\right),\text{ (15.78)}$

with risk-neutral drift ${\tilde{\alpha }}_F\left(t,T\right)={\sigma }_F\left(t,T\right){\displaystyle {\int }_t^T{\sigma }_F\left(t,u\right)\text{d}u.}$ This is necessarily the form for the drift of the forward price process under the risk-neutral measure $\tilde{\mathbb{P}}$.

15.3.2 Relationship between HJM and Affine Yield Models

We can formulate any one-factor short-rate model within the HJM framework. Consider an affine term structure model. The short rate process follows the SDE (15.40) under $\tilde{\mathbb{P}}$ with coefficients given by (15.41):

$\text{d}r\left(t\right)=\left(a_0\left(t\right)+a_1\left(t\right)r\left(t\right)\right)\text{d}t+\sqrt{b_0\left(t\right)+b_1\left(t\right)}\text{d}\tilde{W}\left(t\right).$

Note that $b\left(t,r\right)\equiv \sqrt{b_0\left(t\right)+b_1\left(t\right)r}$ defines the diffusion function of the short rate process, where b2(t, r(t)) = b0(t) + b1(t)r(t). The bond price is in the affine form (15.39):

$Z\left(t,T\right)={\text{e}}^{A\left(t,T\right)-r\left(t\right)C\left(t,T\right)},$

where the functions C and A solve the ODEs (15.43) and (15.44), respectively. According to (15.6), the forward rates are

$f\left(t,T\right)=-\frac{\partial }{\partial T}\ln Z\left(t,T\right)=-\frac{\partial A\left(t,T\right)}{\partial T}+r\left(t\right)\frac{\partial C\left(t,T\right)}{\partial T}.$

Applying the Itô formula, where A(t,T) and C(t,T) are nonrandom functions of time t, we obtain the stochastic differential of the forward rate in the form

$\begin{array}{c}\text{d}f\left(t,T\right)=\frac{\partial C\left(t,T\right)}{\partial T}\text{d}r\left(t\right)+r\left(t\right)\frac{{\partial }^{2}C\left(t,T\right)}{\partial t\partial T}\text{d}t-\frac{{\partial }^{2}A\left(t,T\right)}{\partial t\partial T}\text{d}t\\ =\left[\frac{\partial C\left(t,T\right)}{\partial T}\left(a_0\left(t\right)+a_1\left(t\right)r\left(t\right)\right)+r\left(t\right)\frac{{\partial }^{2}C\left(t,T\right)}{\partial t\partial T}-\frac{{\partial }^{2}A\left(t,T\right)}{\partial t\partial T}\right]\text{d}t\\ +\frac{\partial C\left(t,T\right)}{\partial T}b\left(t,r\left(t\right)\right)\text{d}\tilde{W}\left(t\right).\end{array}$

Equating the diffusion term in the above SDE with (15.77) gives

${\sigma }_F\left(t,T\right)=\frac{\partial C\left(t,T\right)}{\partial T}b\left(t,r\left(t\right)\right)=\frac{\partial C\left(t,T\right)}{\partial T}\sqrt{b_0\left(t\right)+b_1\left(t\right)r\left(t\right)}.\text{ (15.79)}$

Equating the drift term with ${\tilde{\alpha }}_F(t,T)$ in (15.77) gives:

$\begin{array}{l}\frac{\partial C\left(t,T\right)}{\partial T}\left(a_0\left(t\right)+a_1\left(t\right)r\left(t\right)\right)+r\left(t\right)\frac{{\partial }^{2}C\left(t,T\right)}{\partial t\partial T}-\frac{{\partial }^{2}A\left(t,T\right)}{\partial t\partial T}\hfill \\ =\frac{\partial C\left(t,T\right)}{\partial T}{b}^2\left(t,r\left(t\right)\right){\displaystyle {\int }_t^T\frac{\partial C\left(t,u\right)}{\partial u}\text{d}u}\hfill \\ =b^2\left(t,r\left(t\right)\right)\frac{\partial C\left(t,T\right)}{\partial T}\left(C\left(t,T\right)-C\left(t,t\right)\right)\hfill \\ =\left(b_0\left(t\right)+b_1\left(t\right)r\left(t\right)\right)\frac{\partial C\left(t,T\right)}{\partial T}C\left(t,T\right).\hfill \end{array}\text{ (15.80)}$

This is essentially the no-arbitrage condition that must be satisfied by any affine yield model for all times t ≤ T.

15.3.2.1 The Ho–Lee Model in the HJM Framework

Suppose that the diffusion coefficient ${\sigma }_{\Phi }(t,T)$ in the SDE (15.61) for forward rates is constant and equal to σ. The function ΣF is then given by

${\displaystyle {\sum }_F\left(t,T\right)}={\displaystyle {\int }_t^T\sigma \text{d}u=\sigma \left(T-t\right)}.$

The SDE (15.77) takes the form

$\text{d}f\left(t,T\right)={\sigma }^2\left(T-t\right)\text{d}t+\sigma \tilde{W}\left(t\right).$

Integrating the above equation w.r.t. time t gives

$f\left(t,T\right)=f\left(0,T\right)+{\sigma }^{2}t\left(T-t/2\right)+\sigma \tilde{W}\left(t\right).\text{ (15.81)}$

Setting T = t gives the short rate (since r(t) = f(t,t))

$r\left(t\right)=f\left(0,t\right)+{\sigma }^2{t}^2/2+\sigma \tilde{W}\left(t\right).$

In the above equation, we recognize the Ho–Lee model (15.45). Isolating the Brownian term in (15.81) in terms of the forward rates and substitution into the last equation gives

$f\left(t,T\right)=r\left(t\right)+f\left(0,T\right)-f\left(0,t\right)+{\sigma }^{2}t\left(T-t\right).\text{ (15.82)}$

As is seen from the above equation, the spot rate r(t) and forward rate f(t,T) are linearly dependent and hence are perfectly correlated.

Using (15.7), and (15.81) for T = u, we find the bond price:

$\begin{array}{l}Z\left(t,T\right)=\exp \left[-{\displaystyle {\int }_t^T\left(f\left(0,u\right)+{\sigma }^{2}t\left(u-t/2\right)+\sigma \tilde{W}\left(t\right)\right)\text{d}u}\right]\hfill \\ =\exp \left[-{\displaystyle {\int }_t^{T}f\left(0,u\right)\text{d}u-\frac{1}{2}{\sigma }^{2}tT\left(T-t\right)-\sigma \left(T-t\right)\tilde{W}\left(t\right)}\right].\hfill \end{array}$

From (15.9), we obtain $-{\displaystyle {\int }_t^{T}f(0,u)\text{ d}u}=\ln \frac{Z(0,T)}{Z(0,t)}$ and then

$Z\left(t,T\right)=\frac{Z\left(0,T\right)}{Z\left(0,t\right)}\exp \left[-\frac{1}{2}{\sigma }^{2}tT\left(T-t\right)-\sigma \left(T-t\right)\tilde{W}\left(t\right)\right].\text{ (15.83)}$

Since $\frac{Z(0,T)}{Z(0,t)}=e^{-f(0;t,T)(T-t)}$ the ield rate is

$y\left(t,T\right)=-\frac{\ln Z\left(t,T\right)}{T-t}=f\left(0;t,T\right)+\frac{{\sigma }^{2}tT}{2}+\sigma \tilde{W}\left(t\right).$

By eliminating the Brownian term using the equation just below (15.81), we express the yield rate in terms of forward rates (where r(t) = f(t,t)):

$y\left(t,T\right)=f\left(t,t\right)-f\left(0,t\right)+f\left(0;t,T\right)+\frac{1}{2}{\sigma }^{2}t\left(T-t\right).\text{ (15.84)}$

15.3.2.2 The Vasiček Model in the HJM Framework

Let us derive the forward rate SDE and verify the no-arbitrage condition in (15.80) for the Vasicek model. The short-rate process (under $\tilde{\mathbb{P}}$) is driven by the SDE (15.48) which we repeat here:

$\text{d}r\left(t\right)=\left(\tilde{\alpha }-\beta r\left(t\right)\right)\text{d}t+\sigma \text{d}\tilde{W}\left(t\right).$

The functions A and C for the bond price are given in (15.52). Using (15.79) and the expression for C(t, T) in (15.52), we obtain the diffusion coefficient of the forward rate:

${\sigma }_F\left(t,T\right)=\sigma \frac{\partial C\left(t,T\right)}{\partial T}=\sigma \frac{\partial }{\partial T}\left(\frac{1}{\beta }\left(1-{\text{e}}^{-\beta \left(T-t\right)}\right)\right)=\sigma {\text{e}}^{-\beta \left(T-t\right)}.\text{ (15.85)}$

Let us verify that (15.80) holds. Using the expression for C(t,T) and (15.51), we have

$\begin{array}{l}\frac{{\partial }^{2}C\left(t,T\right)}{\partial t\partial T}=\beta {\text{e}}^{-\beta \left(T-t\right)},\hfill \\ \frac{{\partial }^{2}A\left(t,T\right)}{\partial t\partial T}=\left(\tilde{\alpha }-\frac{{\sigma }^2}{\beta }\right){\text{e}}^{-\beta \left(T-t\right)}+\frac{{\sigma }^2}{\beta }{\text{e}}^{-2\beta \left(T-t\right)}.\hfill \end{array}$

For any given r(t) = r, the left-hand side of (15.80) is hence given by

$\begin{array}{l}{\text{e}}^{-\beta \left(T-t\right)}\left(\tilde{\alpha }-\beta r\right)+r\beta {\text{e}}^{-\beta \left(T-t\right)}-\tilde{\alpha }{\text{e}}^{-\beta \left(T-t\right)}+\frac{{\sigma }^2}{\beta }{\text{e}}^{-\beta \left(T-t\right)}-\frac{{\sigma }^2}{\beta }{\text{e}}^{-2\beta \left(T-t\right)}\hfill \\ =\frac{{\sigma }^2}{\beta }\left({\text{e}}^{-\beta \left(T-t\right)}-{\text{e}}^{-2\beta \left(T-t\right)}\right),\hfill \end{array}\text{ (15.86)}$

and the right-hand side of (15.80) is

$\frac{{\sigma }^2}{\beta }{\text{e}}^{-\beta \left(T-t\right)}\left(1-{\text{e}}^{-\beta \left(T-t\right)}\right).\text{ (15.87)}$

Hence the expressions in (15.86) and (15.87) are equal, i.e., the Vasiček short-rate model is of the HJM type for which the no-arbitrage condition holds. Using (15.85), the risk-neutral drift of the forward rate is given by

${\tilde{\alpha }}_F\left(t,T\right)\equiv {\sigma }_F\left(t,T\right){\displaystyle {\sum }_F\left(t,T\right)}=\sigma {\text{e}}^{-\beta \left(T-t\right)}{\displaystyle {\int }_t^T\sigma {\text{e}}^{-\beta \left(u-t\right)}}\text{d}u=\frac{{\sigma }^2}{\beta }{\text{e}}^{-\beta \left(T-t\right)}\left(1-{\text{e}}^{-\beta \left(T-t\right)}\right).$

Thus, for the Vasiček model, the forward rates follow the SDE

$\text{d}f\left(t,T\right)=\frac{{\sigma }^2}{\beta }\left({\text{e}}^{-\beta \left(T-t\right)}-{\text{e}}^{-2\beta \left(T-t\right)}\right)\text{d}t+\sigma {\text{e}}^{-\beta \left(T-t\right)}\text{d}\tilde{W}\left(t\right)\text{ (15.88)}$

under the risk-neutral measure $\tilde{\mathbb{P}}$.

15.4 Multifactor Affine Term Structure Models

In the previous two sections, we discussed one-factor models where Brownian motion is the only source of randomness. One-factor short-rate models offer good analytical tractabil-ity. For many models, a closed-form solution for the bond price can be found. However, one-factor models of interest rates have many drawbacks, including the fact that yield rates for different maturities are perfectly correlated. So the term structure of interest rates for a one-factor model is oversimplified. One of the possible solutions is to consider multifactor interest rate models that involve the short rate along with other random parameters. Consider the following example. Let the short rate follow

$\text{d}r\left(t\right)=\alpha \left(\overline{r}-\beta r\left(t\right)\right)\text{d}t+\sigma \text{d}{W}_1\left(t\right).$

The stochastic volatility model assumes that the volatility σ of the short rate process is stochastic. For example, its square (or variance) is governed by a square-root model,

$\text{d}{\sigma }^2\left(t\right)=\left(\gamma -\delta {\sigma }^2\left(t\right)\right)\text{d}t+\xi \sigma \text{d}{W}_2\left(t\right),$

where W1 and W2 are correlated Brownian motions, and α, γ, δ, ξ are constant parameters. So the interest rate volatility σ (t) is included as the second random variable. This approach can be extended by including another additional factor: the stochastic mean level of the short rate $\overline{r}$. As a result, one can construct a three-factor model with three state variables: the short rate r(t), the volatility σ(t), and the mean level $\overline{r}(t)$.

The multifactor approach provides a greater flexibility in modelling the stochastic term structure of interest rates. However, increasing the number of random factors, reduces the analytical tractability of a model. Valuation of fixed income derivatives often relies on efficient numerical methods. Calibration of a multifactor model can also be a challenge. In this section we discuss a general multifactor affine term structure model and provide several examples of two- and three-factor models. These models are popular due to their good analytical tractability.

Consider a stochastic interest rate model with n state variables X1(t), X2(t),..., Xn(t) (or, as an n-by-1 vector, X(t) = [X1(t), X2(t),..., Xn(t)⊤). The model is said to be affine if the zero-coupon bond prices admit the following exponential form:

$Z\left(t,T\right)=\exp \left[A\left(t,T\right)+{\displaystyle \sum _{j=1}^n{c}_j\left(t,T\right)X_j\left(t\right)}\right]=\exp \left[A\left(t,T\right)+C{\left(t,T\right)}^{⊺}X\left(t\right)\right],\text{ (15.89)}$

where C(t,T)⊤ := [c1(t,T), c2(t,T),..., cn(t,T)]. The model is time homogeneous if the state variables Xj(t) are all time-homogeneous processes and A and C are functions of the time to maturity $\tau =T-t$ only. In this case the bond price function takes the form

$Z\left(t,t+\tau \right)=\exp \left[A\left(\tau \right)+C{\left(\tau \right)}^{⊺}X\left(t\right)\right].\text{ (15.90)}$

Hence, the yield rate is

$y\left(t,t+\tau \right)=-\frac{1}{\tau }\left(A\left(\tau \right)+C{\left(\tau \right)}^{⊺}X\left(t\right)\right).$

By taking the limit $\tau \searrow 0$, we obtain the short-rate process

$r\left(t\right)=y\left(t,t\right)=-\frac{\text{d}A}{\text{d}\tau }\left(0\right)-\frac{\text{d}{C}^{⊺}}{\text{d}\tau }\left(0\right)X\left(t\right).$

As was demonstrated in Section 15.2, a single-factor time-homogeneous model admits a bond pricing function in the affine form, if the short-rate process follows an SDE with a linear drift and a linear squared diffusion coefficient,

$\text{d}r\left(t\right)=\left(\alpha +\beta r\left(t\right)\right)\text{d}t+\sqrt{\lambda +\mu r\left(t\right))}\text{d}\tilde{W}\left(t\right).$

Clearly, certain conditions have to be set on the dynamics of the state vector X(t) in order that $Z(t,t+\tau )$ has the form (15.90). Duffie and Kan proved that the SDE for X(t) under $\tilde{\mathbb{P}}$ has to be of the form

$\text{d}X\left(t\right)=\left(\alpha +BX\left(t\right)\right)\text{d}t+\Sigma D\left(X\left(t\right)\right)\text{d}\tilde{W}\left(t\right),t\ge 0,\text{ (15.91)}$

where D is a diagonal matrix with

$\sqrt{{\lambda }_1+{\mu }_1^{⊺}X\left(t\right),}\sqrt{{\lambda }_2+{\mu }_2^{⊺}X\left(t\right),}...,\sqrt{{\lambda }_n+u_n^{⊺}X\left(t\right)}$

on the main diagonal, $\alpha \text{ and }{\mu }_i,i=1,2,\mathrm{...},n$, are constant n-dimensional vectors, $B={\left[{\beta }_{ij}\right]}_{i,j=1}^n\text{and}\Sigma ={\left[{\sigma }_{ij}\right]}_{i.j=1}^n$ are constant n-by-n matrices, and ${\left\{\tilde{W}\left(t\right)\right\}}_{t\ge 0}$ is an n-dimensional Brownian motion under $\tilde{\mathbb{P}}$. Certain conditions on the model parameters are required to ensure that each of the variance processes ${⋋}_i+{\mu }_i^{\top }\text{X}(t),i=1,2,\mathrm{...},n$, remains positive.

15.4.1 Gaussian Multifactor Models

Let us set the coefficient vectors ${\mu }_1,{\mu }_2,\mathrm{...},{\mu }_n$ to be zero. The SDE (15.91) reduces to the Gaussian form

$\text{d}X\left(t\right)=\left(\alpha +BX\left(t\right)\right)\text{d}t+\Sigma \text{d}\tilde{W}\left(t\right),\text{ (15.92)}$

where all parameters and matrices are constant. It can be shown that the distribution of X(t) is multivariate normal.

15.4.2 Equivalent Classes of Affine Models

The number of possible affine models as defined by (15.91) can be quite large. However, by a transformation of variables, a model can be represented in different ways. Dai and Singleton claimed that the models are considered equivalent if they generate identical prices for all contingent claims. Affine models can be classified according to the number of factors and the number of state variables appearing in the volatility matrix D. There are only two equivalent classes of one-factor models: the Vasiček model and the CIR model. When the number of factors is two, we have three equivalent classes, listed below in their canonical forms.

1. The two-factor Vasiček model

   $\begin{array}{l}\text{d}{X}_1\left(t\right)=-{\beta }_{11}{X}_1\left(t\right)\text{d}t+\text{d}{\tilde{W}}_1\left(t\right),\hfill \\ \text{d}{X}_2\left(t\right)=\left(-{\beta }_{21}{X}_1\left(t\right)-{\beta }_{22}{X}_2\left(t\right)\right)\text{d}t+\text{d}{\tilde{W}}_2\left(t\right).\hfill \end{array}\text{ (15.93)}$

2. The two-factor CIR model (for example, the Longstaff–Schwartz model)

   $\begin{array}{l}\text{d}{X}_1\left(t\right)=\left({\mu }_1-{\beta }_{11}{X}_1\left(t\right)-{\beta }_{12}{X}_2\left(t\right)\right)\text{d}t+\sqrt{X_1\left(t\right)}\text{d}{\tilde{W}}_1\left(t\right),\hfill \\ \text{d}{X}_2\left(t\right)=\left({\mu }_2-{\beta }_{21}{X}_1\left(t\right)-{\beta }_{22}{X}_2\left(t\right)\right)\text{d}t+\sqrt{X_2\left(t\right)}\text{d}{\tilde{W}}_2\left(t\right).\hfill \end{array}\text{ (15.94)}$

3. The two-factor stochastic volatility model (for example, the Fong–Vasiček model)

   $\begin{array}{l}\text{d}{X}_1\left(t\right)=\left({\mu }_1-{\beta }_{11}{X}_1\left(t\right)\right)\text{d}t+\sqrt{X_1\left(t\right)}\text{d}{\tilde{W}}_1\left(t\right),\hfill \\ \text{d}{X}_2\left(t\right)=\left({\mu }_2-{\beta }_{21}{X}_1\left(t\right)-{\beta }_{22}{X}_2\left(t\right)\right)\text{d}t+\left(1+{\delta }_{21}\sqrt{X_1\left(t\right)}\right)\text{d}{\tilde{W}}_2\left(t\right).\hfill \end{array}\text{ (15.95)}$

In each of the above models, $\left(\widetilde{W_1}\left(t\right),\widetilde{W_2}\left(t\right)\right)$ is a standard two-dimensional Brownian motion, under the risk-neutral measure $\tilde{\mathbb{P}}$ with bank account as numéraire. The short rate is assumed to be an affine (linear) function of the two factors:

$r\left(t\right)=a_0+a_1{X}_1\left(t\right)+a_2{X}_2\left(t\right).\text{ (15.96)}$

The coefficients a0, a1, a2 are typically assumed to be constants, although they can also be chosen as nonrandom functions of time t. These parameters, along with those arising from the SDE for each model, can be used to calibrate the affine model to the spot yield curve.

In the two-factor Vasiček model, the parameters in (15.93) and (15.96) are assumed to take on real values where β11 > 0, β22 > 0. It is readily shown that the factors X1(t), X2(t) are jointly normal random variables and hence the short rate is a normal random variable as long as a1, a2 are not both zero. Hence, there is a positive probability that r(t) < 0 for any positive time t > 0. In the two-factor CIR model the parameters in (15.94) are chosen such that μ1 ≥ 0, μ2 ≥ 0, β11 > 0, β22 > β12 ≤ 0, β21 ≤ 0. Under these conditions it can be shown that the factors are nonnegative processes. That is, if the factors start with nonnegative values X1(0) ≥ 0, X2(0) ≥ 0, then X1(t) ≥ 0, X2(t) ≥ 0 for all time t ≥ 0 (almost surely). Assuming a nonnegative initial short rate r(0) ≥ 0, together with the conditions that a0 ≥ 0, a1 > 0, a2 > 0 in (15.96), guarantees that r(t) ≥ 0 for all time t ≥ 0.

In the interest of space, we do not present the details for pricing bonds under these two-factor affine models. Rather, we summarize the basic steps that are similar to those given for the single-factor affine models and leave the rest of the details as exercises at the end of this chapter. For any of the above two-factor affine models we have prices of zero-coupon bonds that are driven by a two-dimensional system of SDEs for the vector X(t) = [X1(t), X2(t)]⊤. Hence, the corresponding bond pricing function can be expressed as a function of the calendar time t and the time-t value of the two factors, i.e., the zero-coupon bond price process Z(t, T) = V(t, X1(t), X2(t)), where V = V(t, x1, x2) is a smooth differentiable function of time t and twice differentiable function of the spot values X1(t) = x1, X2(t) = x2. Recalling our analysis in Chapter 11, we see that the pricing function can be determined by applying the (two-dimensional) discounted Feynman-Kac Theorem 11.20. We can simply use Theorem 11.20 to express V(t, x1, x2) as a solution to a PDE for t < T, subject to the terminal condition V(T, x1, x2) = Z(T, T) = 1. For an affine model, the solution necessarily has the form given by (15.89) for n = 2, X1(t) = x1, X2(t) = x2:

$V\left(t,x_1,x_2\right)={\text{e}}^{A\left(t,T\right)+c_1\left(t,T\right)x_1+c_2\left(t,T\right)x_2}.\text{ (15.97)}$

Note that the exponent is linear in the factor variables x1 and x2. This is an extension of the form of the solution in (15.39) to include two factors rather than just the single short rate variable r. For a time-homogeneous model we have V(t, x1, x2) = v(τ, x1, x2), where the coefficients are functions of τ = T – t: A(t, T) = A(τ), c1(t, T) = c1(τ), c2(t, T) = c2(τ). Substitution of the above exponential form for V into the corresponding bond-pricing PDE (for the particular model) leads to a system of first order ordinary differential equations in the functions A(τ), c1(τ), and c2(τ). This system can then be solved subject to the initial conditions: A(0) = c1(0) = c2(0) = 0. Given the solutions for these coefficient functions, the bond pricing function is then given by (15.97) (see Exercise 15.10).

15.5 Pricing Derivatives under Forward Measures

15.5.1 Forward Measures

Taking the zero-coupon bond Z rather than the bank account B as a numèraire asset allows us to simplify the derivative pricing formula (15.12). Let ${\widehat{\mathbb{P}}}_T^{(Z)}$ denote the EMM relative to the numèraire g(t) = Z(t, T). The probability measure ${\widehat{\mathbb{P}}}_T^{(Z)}$ is called the T-forward measure. It is defined so that the process ${\left\{B\left(t\right)/Z\left(t,T\right)\right\}}_{0\le t\le T}$ is a ${\widehat{\mathbb{P}}}_T^{(Z)}$-martingale. The Radon–Nykodim derivative of ${\tilde{\mathbb{P}}}^{\left(B\right)}\equiv \tilde{\mathbb{P}}\text{w}\text{.r}\text{.t}\text{.}{\tilde{\mathbb{P}}}^{\left(g\right)}\equiv {\tilde{\mathbb{P}}}_T^{\left(Z\right)}\equiv \tilde{\mathbb{P}}$ is

$\varrho \equiv \frac{\text{d}\tilde{\mathbb{P}}}{\text{d}\widehat{\mathbb{P}}}=\frac{B\left(T\right)/B\left(0\right)}{Z\left(T,T\right)/Z\left(0,T\right)}=Z\left(0,T\right)B\left(T\right).\text{ (15.98)}$

Note that B(0) = Z(T, T) = 1. The respective Radon–Nykodim derivative process is

${\varrho }_t\equiv {\left(\frac{\text{d}\tilde{\mathbb{P}}}{\text{d}\widehat{\mathbb{P}}}\right)}_t=\frac{B\left(t\right)/B\left(0\right)}{Z\left(t,T\right)/Z\left(0,T\right)}=\frac{Z\left(0,T\right)B\left(t\right)}{Z\left(t,T\right)},0\le t\le T,\text{ (15.99)}$

where ${\varrho }_T=\varrho \text{.}$ We recall, from (13.129) in Chapter 13, that ${\varrho }_t=\varrho \frac{g}{t}\frac{\to B}{},$ where ${\varrho }_t^{B\to g}=1/{\varrho }_t^{g\to B}=1/{\varrho }_t$ corresponds to changing numèraires from the bank account to the zero-coupon bond in this case. Let payoff V(T) with maturity time T be ${ℱ}_T$-measurable and integrable w.r.t. $\tilde{\mathbb{P}}$. Applying the change of numèraire theorem, i.e., as follows immediately by the risk-neutral pricing formula (13.134) with g(t) = Z(t, T) as numèraire asset price, the time-t price of the claim is given by the conditional expectation under the T-forward measure $\widehat{\mathbb{P}}$:

$V\left(t\right)=g\left(t\right){\widehat{\text{E}}}_t\left[\frac{V\left(T\right)}{g\left(T\right)}\right]=Z\left(t,T\right){\widehat{\text{E}}}_t\left[V\left(T\right)\right].\text{ (15.100)}$

The expected value ${\widehat{\text{E}}}_t\left[V\left(T\right)\right]=V\left(t\right)/Z\left(t,T\right)$ is called the forward price at time t of the payoff V(T) with maturity time T. Recall that the forward price makes the time-t value of forward delivery of V(T) at time T zero.

Example 15.1.

Compute the expectation of the future short rate r(T), conditional on ${ℱ}_t,$ under the forward measure $\widehat{\mathbb{P}}\equiv {\widehat{\mathbb{P}}}_T^{\left(Z\right)}$.

Solution. We note that the Radon–Nykodim derivative process of $\widehat{\mathbb{P}}\equiv {\widehat{\mathbb{P}}}_T^{\left(Z\right)}$ w.r.t. $\widehat{\mathbb{P}}$ is ${\left(\frac{\text{d}\widehat{\mathbb{P}}}{\text{d}\widehat{\mathbb{P}}}\right)}_t=\frac{1}{{\varrho }_t}$. Hence by (15.99),

$\frac{{\left(\frac{\text{d}\widehat{\mathbb{P}}}{\text{d}\tilde{\mathbb{P}}}\right)}_T}{{\left(\frac{\text{d}\widehat{\mathbb{P}}}{\text{d}\tilde{\mathbb{P}}}\right)}_t}=\frac{1/{\varrho }_T}{1/{\varrho }_t}=\frac{{\varrho }_t}{{\varrho }_T}=\frac{1}{Z\left(t,T\right)}\frac{B\left(t\right)}{B\left(T\right)}.$

Thus, we have (by the property(11.83) where r(T) is ${ℱ}_T$-measurable),

$\begin{array}{l}{\widehat{\text{E}}}_t\left[r\left(T\right)\right]={\tilde{\text{E}}}_t\left[\frac{1}{Z\left(t,T\right)}\frac{B\left(t\right)}{B\left(T\right)}r\left(T\right)\right]=\frac{1}{Z\left(t,T\right)}{\tilde{\text{E}}}_t\left[\exp \left(-{\displaystyle {\int }_t^{T}r\left(u\right)\text{d}u}\right)r\left(T\right)\right]\hfill \\ =\frac{1}{Z\left(t,T\right)}{\tilde{\text{E}}}_t\left[-\frac{\partial }{\partial T}\exp \left(-{\displaystyle {\int }_t^{T}r\left(u\right)\text{d}u}\right)\right]\hfill \end{array}$

(assume that we can interchange the operations of differentiation and integration)

$=-\frac{1}{Z\left(t,T\right)}\frac{\partial }{\partial T}\left\{{\tilde{\text{E}}}_t\left[\exp \left(-{\displaystyle {\int }_t^{T}r\left(u\right)\text{d}u}\right)\right]\right\}=-\frac{1}{Z\left(t,T\right)}\frac{\partial Z\left(t,T\right)}{\partial T},$

where the bond price formula (15.11) is used. Using (15.6), we have

$-\frac{1}{Z\left(t,T\right)}\frac{\partial Z\left(t,T\right)}{\partial T}=-\frac{\partial \ln Z\left(t,T\right)}{\partial T}=f\left(t,T\right).$

Hence, the instantaneous forward rate f(t, T) is equal to the mathematical expectation of the short rate r(T) ≡ f(T, T) conditional on ${ℱ}_t$ under the T-forward measure:

${\widehat{\text{E}}}_t\left[r\left(T\right)\right]={\widehat{\text{E}}}_t\left[f\left(T,T\right)\right]=f\left(t,T\right)$

Thus, we conclude that the forward rate process ${\left\{f\left(t,T\right)\right\}}_{0\le t\le T}$ is a ${\widehat{\mathbb{P}}}_T^{\left(Z\right)}$-martingale.

Let the dynamics of the T-maturity bond price be governed by the SDE (15.17) which we repeat here:

$\frac{\text{d}Z\left(t,T\right)}{Z\left(t,T\right)}=r\left(t\right)\text{d}t+{\sigma }_Z\left(t,T\right)\text{d}\tilde{W}\left(t\right),$

where $\tilde{W}$ is a $\tilde{\mathbb{P}}$-Brownian motion. Under the HJM framework with forward rates following the SDE (15.61), we have the SDE (15.76). Hence, equating the volatility terms gives

${\sigma }_Z\left(t,T\right)=-{\Sigma }_F\left(t,T\right)=-{\displaystyle {\int }_t^T{\sigma }_F\left(t,u\right)\text{d}u.}\text{ (15.101)}$

The bond price is given by (15.18). We rewrite it as follows:

$Z\left(t,T\right)=Z\left(0,T\right)B\left(t\right)\exp \left(-\frac{1}{2}{\displaystyle {\int }_0^t{\sigma }_Z^2\left(s,T\right)\text{d}s+{\displaystyle {\int }_0^t{\sigma }_Z\left(s,T\right)\text{d}\tilde{W}\left(s\right)}}\right).\text{ (15.102)}$

The measure $\widehat{\mathbb{P}}\equiv {\widehat{\mathbb{P}}}_T^{\left(Z\right)}$ is defined by the Radon–Nykodim derivative $\frac{\text{d}\widehat{\mathbb{P}}}{\text{d}\widehat{\mathbb{P}}}={\widehat{\varrho }}_T$, where

${\widehat{\varrho }}_t:=\frac{1}{{\varrho }_t}=\frac{Z\left(t,T\right)}{B\left(t\right)Z\left(0,T\right)},0\le t\le T$

with $\varrho t$ given in (15.99). Using the bond price solution (15.102), we have

${\widehat{\varrho }}_t\equiv {\left(\frac{\text{d}\widehat{\mathbb{P}}}{\text{d}\tilde{\mathbb{P}}}\right)}_t=\exp \left(-\frac{1}{2}{\displaystyle {\int }_0^t{\sigma }_Z^2}\left(s,T\right)\text{d}s+{\displaystyle {\int }_0^t{\sigma }_Z\left(s,T\right)\text{d}\tilde{W}\left(s\right)}\right).$

Note that this is precisely the definition of the change of measure as given by the exponential $\tilde{\mathbb{P}}$-martingale in (13.118) since ${\widehat{\varrho }}_t={\varrho }_t^{B\to g}$ with g(t) = Z(t, T). Since we are assuming that asset prices are driven by a single Brownian component, the volatility vector of the numèraire asset is now simply a scalar, σ(g)(t) ≡ σZ(t, T) for fixed maturity T. By Girsanov’s Theorem 11.13 the process

$\widehat{W}\left(t\right):=\tilde{W}\left(t\right)-{\displaystyle {\int }_0^t{\sigma }_Z\left(s,T\right)}\text{d}s,0\le t\le T$

is a standard $\widehat{\mathbb{P}}$-Brownian motion. Using

$\text{d}\tilde{W}\left(t\right)=\text{d}\widehat{W}\left(t\right)+{\sigma }_Z\left(t,T\right)\text{d}t,\text{ (15.1031)}$

we have the following SDE for Z(t, T) under the T-forward measure $\widehat{\mathbb{P}}$:

$\frac{\text{d}Z\left(t,T\right)}{Z\left(t,T\right)}=\left(r\left(t\right)+{\sigma }_Z^2\left(t,T\right)\right)\text{d}t+{\sigma }_Z\left(t,T\right)\text{d}\widehat{W}\left(t\right).$

Within the HJM framework, we hence obtain

$\begin{array}{c}\text{d}f\left(t,T\right)={\sigma }_F\left(t,T\right){\Sigma }_F\left(t,T\right)\text{d}t+{\sigma }_F\left(t,T\right)\text{d}\tilde{W}\left(t\right)\\ ={\sigma }_F\left(t,T\right){\Sigma }_F\left(t,T\right)\text{d}t+{\sigma }_F\left(t,T\right)\left(\text{d}\widehat{W}\left(t\right)-{\Sigma }_F\left(t,T\right)\text{d}t\right)\\ ={\sigma }_F\left(t,T\right)\text{d}\widehat{W}\left(t\right).\end{array}\text{ (15.104)}$

Thus, the forward rate satisfies the driftless SDE df(t, T) = σF(t, T) $\text{d}\widehat{W}\left(t\right)$ and is hence a $\widehat{\mathbb{P}}$-martingale. We arrived at the same conclusion in Example 15.1.

15.5.2 Pricing Stock Options under Stochastic Interest Rates

In this section we present a generalized Black–Scholes formula for option prices under an asset price model with stochastic interest rates. Consider a risky asset such as a stock without dividends. Let the price dynamics of a nondividend paying stock S(t) and the bond price Z(t, T) at time t be governed by

$\frac{\text{d}S\left(t\right)}{S\left(t\right)}=r\left(t\right)\text{d}t+{\sigma }_S\left(t\right)\text{d}\tilde{W}\left(t\right),\text{ (15.105)}$

$\frac{\text{d}Z\left(t,T\right)}{Z\left(t,T\right)}=r\left(t\right)\text{d}t+{\sigma }_Z\left(t,T\right)\text{d}\tilde{W}\left(t\right),\text{ (15.106)}$

with respective log-volatilities σS(t) and σZ(t, T) and where $\tilde{W}$ is a standard (one-dimensional) $\tilde{\mathbb{P}}$-Brownian motion. For a given maturity T, we define

$F_S\left(t\right)=\frac{S\left(t\right)}{Z\left(t,T\right)},0\le t\le T,$

which is the time-t price of the T-maturity forward of the stock. By the definition of the T-forward measure $\widehat{\mathbb{P}}\equiv {\widehat{\mathbb{P}}}_T^{\left(Z\right)}$, all (domestic) nondividend paying asset prices divided by the bond price Z(t, T) are $\widehat{\mathbb{P}}$-martingales. That is, the forward price of the stock is a $\widehat{\mathbb{P}}$-martingale (see (13.128) where A(t) = S(t), g(t) = Z(t, T)). This is also easily shown by computing the stochastic differential of FS(t) expressed in terms of the Brownian increment $\text{d}\widehat{W}\left(t\right)$ (see Exercise 15.8):

$\frac{\text{d}{F}_S\left(t\right)}{F_S\left(t\right)}=\left({\sigma }_S\left(t\right)-{\sigma }_Z\left(t,T\right)\right)\text{d}\widehat{W}\left(t\right)={\sigma }_{F_S}\left(t\right)\text{d}\widehat{W}\left(t\right),\text{ (15.107)}$

where σFS (t) ≡ σS(t)−σZ(t, T) is the log-volatility of FS(t) and $\tilde{W}$ is a standard $\widehat{\mathbb{P}}$-Brownian motion. For the sake of analytical tractability, we now assume σS(t) and σZ(t, T) to be nonrandom (deterministic) continuous functions of time. Using the strong solution to the above SDE, i.e., $F_S(t)=F_S(0){ℰ}_t({\sigma }_{F_S}\cdot \widehat{W})$, we have

$\begin{array}{l}{F}_S\left(T\right)=F_S\left(t\right)\exp \left[-\frac{1}{2}{\displaystyle {\int }_t^T{\left({\sigma }_S\left(u\right)-{\sigma }_Z\left(u,T\right)\right)}^2\text{d}u+}{\displaystyle {\int }_t^T\left({\sigma }_S\left(u\right)-{\sigma }_Z\left(u,T\right)\right)\text{d}\widehat{W}\left(u\right)}\right]\hfill \\ \stackrel{d}{=}{F}_S\left(t\right)\exp \left[-\frac{1}{2}{\overline{\sigma }}_{t,T}^2\left(T-t\right)+{\overline{\sigma }}_{t,T}\sqrt{T-t}\widehat{Z}\right],\hfill \end{array}\text{ (15.108)}$

where ${\widehat{\sigma }}_{t,T}^2:=\frac{1}{T-t}{\displaystyle {\int }_t^T{\left({\sigma }_S\left(u\right)-{\sigma }_Z\left(u,T\right)\right)}^2}$ du, and $\widehat{Z}$ ~ Norm(0, 1) (under measure $\widehat{\mathbb{P}}$) is independent of FS(t).

According to (15.100), the time-t price of a standard European call on the stock with maturity T and strike K is

$C\left(t\right)=Z\left(t,T\right){\widehat{E}}_t\left[{\left(S\left(T\right)-K\right)}^{+}\right].$

Since S(T) = S(T)/Z(T, T) = FS(T), we have

$\begin{array}{l}C\left(t\right)=Z\left(t,T\right){\widehat{E}}_t\left[{\left(F_S\left(T\right)-K\right)}^{+}\right].\hfill \\ =Z\left(t,T\right)\widehat{\text{E}}\left[{\left(F_S\left(t\right){\text{e}}^{-\frac{1}{2}{\overline{\sigma }}_{t,T}^2\left(T-t\right)+{\overline{\sigma }}_{t,T}\sqrt{T-t}\widehat{Z}}-K\right)}^{+}|F_S\left(t\right)\right]\hfill \end{array}\text{ (15.109)}$

where the expectation is conditional on the time-t forward price FS(t) = S(t)/Z(t, T). Note that, since FS(t) is independent of $\widehat{Z}$, this expectation is computed by making use of the independence proposition. In particular, the expectation is the same as that of a standard call with strike K, zero “effective interest rate and dividend on the underlying”, spot value FS(t), volatility ${\overline{\sigma }}_{t,T}$, and time to maturity T − t. Hence, we obtain the well-known Black formula for the value of a call option on a stock:

$\begin{array}{l}C\left(t\right)=Z\left(t,T\right)\left[F_S\left(t\right)\mathcal{N}\left(d_{+}\left(t\right)\right)-K\mathcal{N}\left(d_{-}\left(t\right)\right)\right]\hfill \\ =S\left(t\right)\mathcal{N}\left(d_{+}\left(t\right)\right)-KZ\left(t,T\right)\mathcal{N}\left(d_{-}\left(t\right)\right)\hfill \end{array}\text{ (15.110)}$

where

$d_{\pm }\left(t\right)=\frac{1}{{\overline{\sigma }}_{t,T}\sqrt{T-t}}\left[\ln \left(\frac{S\left(t\right)}{KZ\left(t,T\right)}\right)\pm \frac{1}{2}{\overline{\sigma }}_{t,T}^2\left(T-t\right)\right].$

Note that here we have expressed the time-t call price in terms of the time-t stock price and zero-coupon bond price S(t) and Z(t, T). Plugging in the time-t spot values for the stock and the bond gives the pricing function for the call. The above formulation is readily extended to the case with multiple stocks driven by a vector Brownian motion where the stocks are correlated with each other as well as being correlated with the bond price process (see Exercise 15.9).

15.5.3 Pricing Options on Zero-Coupon Bonds

The risk-neutral pricing formulae (15.12) and (15.100) allow for computing no-arbitrage prices of any attainable claim. We assume that an EMM (risk-neutral measure) exists, implying the absence of arbitrage. Consider a European-style claim with maturity T on a zero-coupon bond maturing at time T′ > T. The payoff function V(T) of such a claim has the form $\Lambda \left(Z\left(T,T\prime \right)\right)$. The no-arbitrage value $V\left(t\right),t\le T$, of the claim is given by

$V\left(t\right)={\tilde{\text{E}}}_t\left[D\left(t,T\right)\Lambda \left(Z\left(T,T^{\prime }\right)\right)\right]={\tilde{\text{E}}}_t\left[\exp \left(-{\displaystyle {\int }_t^{T}r\left(u\right)\text{d}u}\right)\Lambda \left(Z\left(T,T^{\prime }\right)\right)\right]\text{ (15.111)}$

under the risk-neutral measure $\tilde{\mathbb{P}}\equiv {\tilde{\mathbb{P}}}^{\left(B\right)}$, or

$V\left(t\right)=Z\left(t,T\right){\widehat{\text{E}}}_t\left[\Lambda \left(Z\left(T,T^{\prime }\right)\right)\right]\text{ (15.112)}$

when the equivalent T-forward measure $\widehat{\mathbb{P}}\equiv {\widehat{\mathbb{P}}}_T^{\left(Z\right)}$ is used. The main drawback of the pricing formula (15.111) is that it is necessary to find the joint distribution of ${\displaystyle {\int }_r^{T}r\left(u\right)}$ du and Z(T, T′) under $\tilde{\mathbb{P}}$ to find the expectation. When using (15.112), we only need to find the dynamics of the bond price Z(t, T′) under the T-forward measure.

Let FZ(t) ≡ FZ(t; T, T′) denote the T-forward price of the T′-maturity bond at time t:

$F_Z\left(t\right)=\frac{Z\left(t,T^{\prime }\right)}{Z\left(t,T\right)},0\le t\le T\le T^{\prime }.$

Note that the bond price Z(t, T′) is a domestic (nondividend paying) asset. Hence, under the risk-neutral measure $\tilde{\mathbb{P}}$, its price follows the SDE in (15.106) (now with maturity T′ replacing T),

$\text{d}Z\left(t,T^{\prime }\right)=Z\left(t,T^{\prime }\right)\left[r\left(t\right)\text{d}t+{\sigma }_Z\left(t,T^{\prime }\right)\text{d}\tilde{W}\left(t\right)\right].$

In the T-forward measure, with g(t) = Z(t, T) as numèraire asset price, the forward price FZ(t) is a $\widehat{\mathbb{P}}$-martingale satisfying a driftless SDE,

$\frac{\text{d}{F}_Z\left(t\right)}{F_Z\left(t\right)}=\left({\sigma }_Z\left(t,T^{\prime }\right)-{\sigma }_Z\left(t,T\right)\right)\text{d}\widehat{W}\left(t\right).\text{ (15.113)}$

The reader will recognize this as an application of (13.128) in Chapter 13, where A(t) = Z(t, T′), g(t) = Z(t, T). Since the forward price FZ(t) is a ${\widehat{\mathbb{P}}}_T^{\left(Z\right)}$-martingale, we have $\widehat{\text{E}}$ t [FZ(T)] = FZ(t). By the above definition we also have $F_z\left(T\right)=\frac{Z\left(T,T\text{'}\right)}{Z\left(T,T\right)}=Z\left(T,T\text{'}\right)$, which gives

${\widehat{\text{E}}}_t\left[Z\left(T,T^{\prime }\right)\right]=F_z\left(t\right).\text{ (15.114)}$

In other words, FZ(t) is the time-t forward price for delivery of Z(T, T′) at time T.

Alternatively, to find the dynamics of FZ(t), we use the solution (15.18) for deducing a relation between bond prices with maturities T and T′:

$F_Z\left(t\right)=\frac{Z\left(t,T^{\prime }\right)}{Z\left(t,T\right)}=\frac{Z\left(0,T^{\prime }\right)}{Z\left(0,T\right)}{\text{e}}^{-\frac{1}{2}{\displaystyle {\int }_0^t\left({\sigma }_Z^2\left(s,T^{\prime }\right)-{\sigma }_Z^2\left(s,T\right)\right)\text{d}s+{\displaystyle {\int }_0^t\left({\sigma }_Z\left(s,T^{\prime }\right)-{\sigma }_Z\left(s,T\right)\right)\text{d}\tilde{W}\left(s\right)}}}\text{ (15.115)}$

(now we use (15.103) to change the probability measure)

$\begin{array}{l}=\frac{Z\left(0,T^{\prime }\right)}{Z\left(0,T\right)}{\text{e}}^{-\frac{1}{2}{\displaystyle {\int }_0^t\left({\sigma }_Z^2\left(s,T^{\prime }\right)-2{\sigma }_Z\left(s,T^{\prime }\right)\right){\sigma }_Z\left(s,T\right)+{\sigma }_Z^2\left(s,T\right))\text{d}s+{\displaystyle {\int }_0^t\left({\sigma }_Z\left(s,T^{\prime }\right)-{\sigma }_Z\left(s,T\right)\right)\text{d}\widehat{W}\left(s\right)}}}\hfill \\ =\frac{Z\left(0,T^{\prime }\right)}{Z\left(0,T\right)}{\text{e}}^{-\frac{1}{2}{\displaystyle {\int }_0^t{\left({\sigma }_Z\left(s,T^{\prime }\right)-{\sigma }_Z\left(s,T\right)\right)}^2\text{d}s+{\displaystyle {\int }_0^t\left({\sigma }_Z\left(s,T^{\prime }\right)-{\sigma }_Z\left(s,T\right)\right)\text{d}\widehat{W}\left(s\right)}}}.\hfill \end{array}\text{ (15.116)}$

On the right hand side of (15.116) we recognize a stochastic exponential. Hence, (15.113) follows. In particular, we can relate the forward price of the T′-maturity bond at time t to that at time T by simply dividing (15.116) into the same expression for t = T:

$\frac{F_Z\left(T\right)}{F_Z\left(t\right)}=\frac{Z\left(T,T^{\prime }\right)}{F_Z\left(t\right)}={\text{e}}^{-\frac{1}{2}{\displaystyle {\int }_t^T{\left({\sigma }_Z\left(s,T^{\prime }\right)-{\sigma }_Z\left(s,T\right)\right)}^2\text{d}s+{\displaystyle {\int }_t^T\left({\sigma }_Z\left(s,T^{\prime }\right)-{\sigma }_Z\left(s,T\right)\right)\text{d}\widehat{W}\left(s\right)}}}.\text{ (15.117)}$

The reader will note that this also follows by directly integrating (15.113).

Let us consider pricing standard European call and put options on the bond. For example, the call option with maturity T and strike K on the bond Z(T, T′) gives the right to buy the bond at time T for K. For some term structure models, such as the Gaussian HJM model, one is able to derive the pricing formulae for standard European options in closed form.

Suppose that the bond volatility function σZ(t, T) is a deterministic (nonrandom) function. Under the HJM framework we have the relation (15.101), which implies that the diffusion coefficient of the forward rate in the SDE (15.61) is also a deterministic function of time t. It follows that the drift and diffusion coefficients in both (15.77) and (15.104) are nonrandom functions of t. Hence, the forward rate is a Gaussian process in either measure $\tilde{\mathbb{P}}$ or $\widehat{\mathbb{P}}$. We see from 15.117 that $\frac{F_Z\left(T\right)}{F_Z\left(t\right)}$ is a log-normal random variable. In particular, taking logarithms on both sides of (15.117) gives

$\begin{array}{l}\ln Z\left(T,T^{\prime }\right)=\ln F_Z\left(t\right)-\frac{1}{2}{\displaystyle {\int }_t^T{\left({\sigma }_Z\left(s,T^{\prime }\right)-{\sigma }_Z\left(s,T\right)\right)}^2\text{d}s+}{\displaystyle {\int }_t^T\left({\sigma }_Z\left(s,T^{\prime }\right)-{\sigma }_Z\left(s,T\right)\right)\text{d}\widehat{W}\left(s\right)}\hfill \\ \stackrel{d}{=}\ln F_Z\left(t\right)-\frac{1}{2}{\overline{\sigma }}^2\left(T-t\right)+\overline{\sigma }\sqrt{T-t}\widehat{Z}\hfill \end{array}\text{ (15.118)}$

where $\widehat{Z}$ ~ Norm(0, 1) (under $\widehat{\mathbb{P}}$) and we conveniently define the constant (for given t, T, T′)

${\overline{\sigma }}^2\equiv {\overline{\sigma }}_{t,T,T^{\prime }}^2:=\frac{1}{T-t}{\displaystyle {\int }_t^T{\left({\sigma }_Z\left(s,T^{\prime }\right)-{\sigma }_Z\left(s,T\right)\right)}^2\text{d}s.}$

This corresponds to the time-averaged square difference of the volatilities of the bond price with respective maturities T′ and T. Note that the last line in (15.118) is an equality in distribution where we used the fact that the Ito integral is a normal random variable with zero mean and variance ${\overline{\sigma }}^2\left(T-t\right)$. The reader can readily verify that this follows by Itô isometry, since the integrand is a nonrandom square-integrable function of s (for fixed T, T′). Moreover, the above Itô integral is independent of ${ℱ}_t,\text{i}\text{.e}\text{.,}\widehat{Z}$ is independent of ${ℱ}_t$ (and, of course, also independent of FZ(t)).

Based on the above properties, we can now readily price a call option with maturity T and strike K, which is written on the underlying bond with value Z(T, T′). The price of the call at time t is given by computing the conditional expectation in (15.112). The steps are exactly as those for computing the price of a standard European call. We condition on ${ℱ}_t$, where FZ(t) is ${ℱ}_t$-measurable and independent of $\widehat{Z}$. According to (15.118), we substitute $Z\left(T,T\prime \right)=F_Z\left(t\right){\text{e}}^{-\frac{1}{2}{\sigma }^2\left(T-t\right)+\overline{\sigma }\sqrt{T-t}\widehat{Z}}$ into the conditional expectation, and we use the independence proposition to reduce the calculation to an unconditional expectation. The latter is then computed by the usual expectation identities in the Appendix (or by simply recognizing the call pricing function at hand in the second line below):

$\begin{array}{c}C\left(t\right)=Z\left(t,T\right)\widehat{\text{E}}\left[{\left(Z\left(T,T^{\prime }\right)-K\right)}^{+}|{ℱ}_t\right]\\ =Z\left(t,T\right)\widehat{\text{E}}\left[{\left(F_Z\left(t\right){\text{e}}^{-\frac{1}{2}{\overline{\sigma }}^2\left(T-t\right)+\overline{\sigma }\sqrt{T-t}\widehat{Z}}-K\right)}^{+}|{ℱ}_t\right]\\ =Z\left(t,T\right)\widehat{\text{E}}\left[{\left(x{\text{e}}^{-\frac{1}{2}{\overline{\sigma }}^2\left(T-t\right)+\overline{\sigma }\sqrt{T-t}\widehat{Z}}-K\right)}^{+}\right]{|}_{x=F_Z\left(t\right)}\\ =Z\left(t,T\right)\left[F_Z\left(t\right)\mathcal{N}\left(d_{+}\left(t\right)\right)-K\mathcal{N}\left(d_{-}\left(t\right)\right)\right]\\ =Z\left(t,T^{\prime }\right)\mathcal{N}\left(d_{+}\left(t\right)\right)-KZ\left(t,T\right)\mathcal{N}\left(d_{-}\left(t\right)\right),\end{array}\text{ (15.119)}$

where

$d_{\pm }\left(t\right):=\frac{1}{\overline{\sigma }\sqrt{T-t}}\left[\ln \left(\frac{Z\left(t,T^{\prime }\right)}{KZ\left(t,T\right)}\right)\pm \frac{1}{2}{\overline{\sigma }}^2\left(T-t\right)\right].$

Note: Z(t, T)FZ(t) = Z(t, T′) and $\frac{F_Z(t)}{K}=\frac{Z\left(t,T\prime \right)}{KZ\left(t,T\right)}$.

15.6 LIBOR Model

15.6.1 LIBOR Rates

LIBOR, which stands for London InterBank Offer Rate, refers to the market interest rate. Let L(t; T, T + τ) denote an annual simple rate of interest that is locked at time t for borrowing from time T to time T + τ, where 0 ≤ t T ≤ T + τ. That is, $1 invested at time T will grow to 1 + τL(t; T, T + τ) dollars at time T + τ. We call L(t; T, T + τ) the forward LIBOR.

In Section 15.1.2 we discussed how the rate on a loan for a period between times T and T′ = T + τ can be locked in at time t. We purchase one zero-coupon bond maturing at time T and finance this purchase by selling short $\frac{Z(t,T)}{Z\left(t,T+\tau \right)}$ units of the bond maturing at time T + τ. The cost of setting up this portfolio is zero. The forward LIBOR rate, which is the effective simple rate of interest applied over the interval [T, T + τ], is calculated as follows:

$1+\tau L\left(t;T,T+\tau \right)=\frac{Z\left(t,T\right)}{Z\left(t,T+\tau \right)}\Rightarrow L\left(t;T,T+\tau \right)=\frac{1}{\tau }\left[\frac{Z\left(t,T\right)}{Z\left(t,T+\tau \right)}-1\right].\text{ (15.120)}$

The interest period τ is often referred as the tenor, and it is typically equal to 0.25 for a three-month LIBOR or 0.5 for a six-month LIBOR. If t = T, then we call L(T; T, T + τ) the spot LIBOR. It is given by

$L\left(T;T,T+\tau \right)=\frac{1}{\tau }\left[\frac{1}{Z\left(T,T+\tau \right)}-1\right].$

15.6.2 Brace–Gatarek–Musiela Model of LIBOR Rates

Pricing interest rate derivatives such as caps and swaps requires a tractable model of floating rates. To adapt the Black–Scholes formula for stock options to the case with fixed-income products, it is desirable to assume that the risk-neutral dynamics of floating rates is log-normal. Suppose that caps and swaps are written on forward rates f(t,T). It can be shown that if the diffusion coefficient in the SDE (15.61) is proportional to the forward rate, i.e., if σF(t,T) = σ(t,T)f(t,T) with σ(t,T) assumed to be a nonrandom function, then the forward rates governed by the risk-neutral SDE (15.77) can explode in finite time. Such behaviour of forward rates is caused by the risk-neutral drift term $\tilde{\alpha }\left(t,T\right)$ in (15.77), which takes the form

$\tilde{\alpha }\left(t,T\right)=\sigma \left(t,T\right)f\left(t,T\right){\displaystyle {\int }_t^T\sigma \left(t,u\right)f\left(t,u\right)\text{d}u.}$

To overcome this problem, Brace, Gatarek, and Musiela (BGM) have suggested using the LIBOR forward rates L(t; T, T+τ), which are simple rates of interest, instead of continuously compounding forward rates.

Let us consider the case with a single maturity T ≤ T* and tenor τ > 0. For notational simplicity we write L(t) ≡ L(t; T, T + τ). Here we present the theory for LIBOR forward rates driven by a single Brownian factor, although the extension to multiple Brownian factors follows readily. Hence, suppose that the bond price process is governed by the SDE in the form of (15.17). As follows from (15.113), the (T + τ)-forward price of the bond, $\frac{Z\left(t,T\right)}{Z\left(t,T+\tau \right)}$, is a $\widehat{\mathbb{P}}$-martingale, or all 0 ≤ t ≤ T, where $\widehat{\mathbb{P}}\equiv {\widehat{\mathbb{P}}}_{T+\tau }^{\left(Z\right)}$ is the (T + τ)-forward measure. From (15.120), we have that the LIBOR rate L(t) is a strictly positive $\widehat{\mathbb{P}}$-martingale as well. According to the Brownian martingale representation theorem 11.14, there exists an $\mathbb{F}$-adapted process v(t, T) such that

$\text{d}L\left(t\right)=\upsilon \left(t,T\right)L\left(t\right)\text{d}\widehat{W}\left(t\right),0\le t\le T,\text{ (15.121)}$

where $\tilde{W}$ is a $\widehat{\mathbb{P}}$-Brownian motion. The process v(t, T) relates to the zero-coupon volatilities as follows:

$\upsilon \left(t,T\right)=\left(\frac{1+\tau L\left(t\right)}{\tau L\left(t\right)}\right)\left[{\sigma }_Z\left(t,T\right)-{\sigma }_Z\left(t,T+\tau \right)\right].\text{ (15.122)}$

The proof of (15.122) is left as an exercise at the end of this chapter. Notice that $v\left(t,T\right)\equiv {\sigma }_L^{\left(\tau \right)}\left(t,T\right)$ is the log-volatility of the process L(t; T, T + τ) for 0 ≤ t ≤ T.

The Brace–Gatarek–Musiela (BGM) model for forward LIBOR rates is constructed such that v(t, T) is a deterministic (nonrandom) function of time for 0 ≤ t ≤ T. As a result, the forward LIBOR rate L(T) conditional on L(t) has a log-normal distribution under measure $\widehat{\mathbb{P}}$ with conditional mean and variance:

${\widehat{\text{E}}}_t\left[\ln L\left(T\right)\right]=\ln L\left(t\right)-\frac{1}{2}\left(T-t\right){\overline{\upsilon }}_{t,T,}^2\text{ (15.123)}$

$\widehat{\text{Var}}\left(\ln L\left(T\right)\right)={\overline{\upsilon }}_{t,T}^2\left(T-t\right)\text{ (15.124)}$

with time-averaged variance ${\overline{v}}_{t,T}^2:=\frac{1}{T-t}{\displaystyle {\int }_t^T{v}^2\left(s,T\right)\text{d}s}$. These expressions follow readily by simply using the solution to the SDE (15.121) as an exponential $\widehat{\mathbb{P}}$-martingale for L(t) at time t and L(T) at time T,

$L\left(T\right)=L\left(t\right){\text{e}}^{-\frac{1}{2}{\displaystyle {\int }_t^T{\upsilon }^2}\left(s,T\right)\text{d}s+{\displaystyle {\int }_t^T\upsilon }\left(s,T\right)\text{d}\widehat{W}\left(s\right)}\stackrel{d}{=}L\left(t\right){\text{e}}^{-\frac{1}{2}\left(T-t\right){\overline{\upsilon }}_{t,T}^2+{\overline{\upsilon }}_{t,T}\sqrt{T-t}\widehat{Z}}.\text{ (15.125)}$

The second equality is in distribution where ${\displaystyle {\int }_t^{T}v}\left(s,T\right)\text{d}\widehat{W}\left(s\right)~Norm\left(0,{\overline{v}}_{t,T}\sqrt{T-t}\right)$ under $\widehat{\mathbb{P}}$ and $\widehat{Z}$ denotes a standard normal random variable under $\widehat{\mathbb{P}}$. The log-normality of the forward LIBOR rate, i.e., the representation in (15.125), allows for the construction of pricing formulae for caps and swaps in closed form. One of the main results is the Black caplet formula presented in the next subsection.

15.6.3 Pricing Caplets, Caps, and Swaps

A cap is defined as a portfolio of caplets. Hence, to find the no-arbitrage price of a cap, it suffices to find the price of a single caplet, Caplet(t), with 0 ≤ t ≤ T. Under the BGM model, a caplet is a European call on the spot LIBOR rate. Its payoff at maturity T + τ is (L(T) − κ)+τ, where L(T) ≡ L(T; T, T + τ) and κ > 0 is a strike rate. By (15.13), the time-t price of the caplet is

$\begin{array}{c}\text{Caplet}\left(t\right)=\tau {\tilde{\text{E}}}_t\left[D\left(t,T+\tau \right){\left(L\left(T\right)-k\right)}^{+}\right]\\ ={\tilde{\text{E}}}_t\left[\frac{B\left(t\right)}{B\left(T+\tau \right)}{\left(\frac{1}{Z\left(T,T+\tau \right)}-1-\kappa \tau \right)}^{+}\right].\end{array}$

From the bond price formula (15.11), we have

$Z\left(T,T+\tau \right)={\tilde{\text{E}}}_T\left[\frac{B\left(T\right)}{B\left(T+\tau \right)}\right].$

Using the tower property and ${ℱ}_T$-measurability of Z(T, T + τ) gives

$\begin{array}{c}\text{Caplet}\left(t\right)={\tilde{\text{E}}}_t\left[{\tilde{\text{E}}}_T\left[\frac{B\left(t\right)}{B\left(T+\tau \right)}{\left(\frac{1}{Z\left(T,T+\tau \right)}-1-\kappa \tau \right)}^{+}\right]\right]\\ ={\tilde{\text{E}}}_t\left[\frac{B\left(t\right)}{B\left(T\right)}{\left(\frac{1}{Z\left(T,T+\tau \right)}-1-\kappa \tau \right)}^{+}{\tilde{\text{E}}}_T\left[\frac{B\left(T\right)}{B\left(T+\tau \right)}\right]\right]\\ ={\tilde{\text{E}}}_t\left[\frac{B\left(t\right)}{B\left(T\right)}{\left(\frac{1}{Z\left(T,T+\tau \right)}-1-\kappa \tau \right)}^{+}Z\left(T,T+\tau \right)\right]\\ =\left(1+\kappa \tau \right){\tilde{\text{E}}}_t\left[\frac{B\left(t\right)}{B\left(T\right)}{\left(\frac{1}{1+\kappa \tau }-Z\left(T,T+\tau \right)\right)}^{+}\right].\end{array}$

That is, the caplet is equivalent to a put option on the zero-coupon bond Z(T, T + τ) with strike $\frac{1}{1+\kappa \mathcal{T}}$ and maturity T.

The price of a caplet is easier to evaluate by using a forward measure. As was demonstrated in the previous subsection, the forward LIBOR rate L(T) conditional on L(t) has a log-normal distribution under the forward measure ${\widehat{\mathbb{P}}}_{T+\mathcal{T}}^{\left(Z\right)}$. The price of the caplet at time t is

$\text{Caplet}\left(t\right)=\tau Z\left(t,T+\tau \right){\widehat{\text{E}}}_t\left[{\left(L\left(T\right)-\kappa \right)}^{+}\right],\text{ (15.126)}$

where $\widehat{\text{E}}$[·] denotes the expectation under ${\widehat{\mathbb{P}}}_{T+\mathcal{T}}^{\left(Z\right)}$. We leave it to the reader to show (by using similar steps as pricing a standard call with the use of (15.125)) that Black’s caplet formula obtains:

$\text{Caplet}\left(t\right)=\tau Z\left(t,T+\tau \right)\left[L\left(t\right)\mathcal{N}\left(d_{+}\left(t\right)\right)-\kappa \mathcal{N}\left(d_{-}\left(t\right)\right)\right],\text{ (15.127)}$

where

$d_{\pm }\left(t\right)=\frac{1}{{\overline{\upsilon }}_{t,T}\sqrt{T-T}}\left[\ln \frac{L\left(t\right)}{\kappa }\pm \frac{1}{2}{\overline{\upsilon }}_{t,T}^2\left(T-t\right)\right].$

A cap is a series of caplets that pays $\mathcal{T}{\left(L\left(T_{i-1};T_{i-1,}{T}_i\right)-\kappa \right)}^{+}$ at time Ti = T0 + iτ for all i = 1, 2,..., n. The total value of the cap at time t ≤ T0 is equal to the sum of all caplet values. Each i-th caplet is valued by considering the expectation ${\widehat{\text{E}}}_t^{\left(T_i\right)}[\cdot ]$ conditional on ${ℱ}_t$ under the Ti-forward measure ${\widehat{\mathbb{P}}}_{T_i}^{\left(Z\right)}$. Summing each caplet value gives the price of the cap:

$\begin{array}{c}\text{Cap}\left(t\right)={\displaystyle \sum _{i=1}^n\tau Z\left(t,T_i\right)}{\widehat{\text{E}}}_t^{\left(T_i\right)}\left[{\left(L\left(T_{i-1};T_{i-1},T_i\right)-\kappa \right)}^{+}\right]\\ =\tau {\displaystyle \sum _{i=1}^{n}Z\left(t,T_i\right)}\left[L\left(t;T_{i-1},T_i\right)\mathcal{N}\left(d_{+}^{\left(i-1\right)}\left(t\right)\right)-\kappa \mathcal{N}\left(d_{-}^{\left(i-1\right)}\left(t\right)\right)\right]\end{array}\text{ (15.128)}$

where

$d_{\pm }^{\left(i-1\right)}\left(t\right)=\frac{1}{{\overline{\upsilon }}_{t,T_{i-1}}\sqrt{T_{i-1}-t}}\left[\ln \frac{L\left(t;T_{i-1},T_i\right)}{\kappa }\pm \frac{1}{2}{\overline{\upsilon }}_{t,T_{i-1}}^2\left(T_{i-1}-t\right)\right]$

and $v_{t,T_{i-1}}^2=\frac{1}{T_{i-1}-t}{\displaystyle {\int }_t^{T_{i-1}}{v}^2}\left(s,T_{i-1}\right)$ ds.

The price of a payer swap at time-t is also easy to evaluate by using forward measures. The holder of the swap receives τ(L(Ti−1; Ti>−1, Ti) >− κ) at time Ti = T0 + iτ for all i = 1, 2, ..., n. The no-arbitrage price at time t is now readily computed by choosing the Ti-forward measure ${\widehat{\mathbb{P}}}_{T_i}^{\left(Z\right)}$ for each i-th payoff term:

$\text{Swap}\left(t\right)={\displaystyle \sum _{i=1}^n\tau Z\left(t,T_i\right)}{\widehat{\text{E}}}_t^{\left(T_i\right)}\left[L\left(T_{i-1};T_{i-1},T_i\right)-\kappa \right]$

(using the fact that the LIBOR rate L(t; Ti−1, Ti) is a ${\widehat{\mathbb{P}}}_{T_i}^{\left(Z\right)}$-martingale)

$\begin{array}{c}=\tau {\displaystyle \sum _{i=1}^{n}Z\left(t,T_i\right)\left[L\left(t;T_{i-1},T_i\right)-\kappa \right]}={\displaystyle \sum _{i=1}^n\tau Z\left(t,T_i\right)}\left[\frac{Z\left(t,T_{i-1}\right)-Z\left(t,T_i\right)}{\tau Z\left(t,T_i\right)}-\kappa \right]\\ ={\displaystyle \sum _{i=1}^n\left[Z\left(t,T_{i-1}\right)-Z\left(t,T_i\right)\right]-\tau \kappa {\displaystyle \sum _{i=1}^{n}Z\left(t,T_i\right)}}\end{array}\text{ (15.129)}$

$=\left[Z\left(t,T_0\right)-Z\left(t,T_n\right)\right]-\tau \kappa {\displaystyle \sum _{i=1}^{n}Z\left(t,T_i\right)}.\text{ (15.130)}$

In obtaining (15.129) we used the definition in (15.120), i.e., $L\left(t;T_{i-1,}{T}_i\right)=\frac{z\left(t,T_{i-1}\right)-z\left(t,T_i\right)}{\mathcal{T}z\left(t,T_i\right)}$.

The forward swap rate that makes the swap contract have a no-arbitrage price of zero at time t is hence

$\kappa \left(t;T_0,T_n\right)=\frac{Z\left(t,T_0\right)-Z\left(t,T_n\right)}{\tau {\displaystyle {\sum }_{i=1}^{n}Z\left(t,T_i\right)}}.\text{ (15.131)}$

Substituting the expression Z(t,T0) − Z(t,Tn), in terms of this forward swap rate, into (15.130) gives

$\text{Swap}\left(t\right)=\tau \left[\kappa \left(t;T_0,T_n\right)-\kappa \right]{\displaystyle \sum _{i=1}^{n}Z\left(t,T_i\right)}.\text{ (15.132)}$

15.7 Exercises

1. Exercise 15.1. Suppose that the (continuously compounded) spot rates for the next three years are

   $\frac{T123}{y\left(0,T\right)3\%3.25\%3.5\%}$

   Find the forward rates f(0; 1, 2), f(0; 1, 3), and f(0; 2, 3).

2. Exercise 15.2. Consider a derivative which has the payoff (y(T,T+1)−y(T,T+5))+. Why might it be inappropriate to use a one-factor interest rate model to value this derivative?
3. Exercise 15.3. Consider the Hull–White model (15.29). Show that the short rate r(t) is a Gaussian process. Find the mean and variance of r(T) conditional on r(t) for 0 ≤ t ≤ T.

4. Exercise 15.4. The short-rate prices under the Pearson–Sun (PS) model follows the SDE

   $\text{d}r\left(t\right)=\left(\alpha -\beta r\left(t\right)\right)\text{d}t+\sigma \sqrt{r\left(t\right)-\lambda }\text{d}W\left(t\right).$

   Make use of the CIR bond pricing formula to derive bond prices for the PS model.

5. Exercise 15.5. Find the forward rates f(t, T) in the Vasiček model.

6. Exercise 15.6. Consider an extended CIR model where the short rate r(t) follows the SDE

   $\text{d}r\left(t\right)=\left(\alpha \left(t\right)-\beta \left(t\right)r\left(t\right)\right)\text{d}t+\sigma \left(t\right)\sqrt{r\left(t\right)}\text{d}\tilde{W}\left(t\right)$

   under the risk-neutral measure $\tilde{\mathbb{P}}$. Here, α(t), β(t), and σ(t) > 0 are continuous ordinary (nonrandom) functions. Show that the price of a zero-coupon bond is

   $Z\left(t,T\right)=\exp \left(-{\displaystyle {\int }_t^{T}C\left(s,T\right)\alpha \left(s\right)\text{d}s-C\left(t,T\right)r\left(t\right)}\right),t\le T,$

   where C(t, T) solves the first order differential equation

   $C^{\prime }\left(t,T\right)=\beta \left(t\right)C\left(t,T\right)+\frac{{\sigma }^2\left(t\right)}{2}{C}^2\left(t,T\right)-1$

   for t < T, subject to C(T, T) = 0.

7. Exercise 15.7. A two-factor HJM model is given by the SDE

   $\text{d}f\left(t,T\right)=\alpha \left(t,T\right)\text{d}t+{\sigma }_1\left(t,T\right)\text{d}{W}_1\left(t\right)+{\sigma }_2\left(t,T\right)\text{d}{W}_2\left(t\right),$

   where W1 and W2 are independent Brownian motions.

   1. (a) Find the SDE for the discounted zero-coupon bond price process $\overline{Z}\left(t,T\right)=D\left(t\right)Z\left(t,T\right)$ and show that

      $\overline{Z}\left(t,T\right)=\overline{Z}\left(0,T\right)\exp \left(-{\displaystyle {\int }_0^{t}A\left(a,T\right)\text{d}s-{\displaystyle {\int }_0^t{\Sigma }_1\left(s,T\right)}\text{d}{W}_1\left(s\right)-{\displaystyle {\int }_0^t{\Sigma }_2\left(s,T\right)}\text{d}{W}_2\left(s\right)}\right),$

      where ${\displaystyle \sum \left(t,T\right)={\displaystyle {\int }_t^T{\sigma }_i\left(t,u\right)\text{d}u\text{for}i=1,2\text{and}A\left(t,T\right)={\displaystyle {\int }_t^T\alpha \left(t,u\right)\text{d}u}}}$.

   2. (b) Show that the no-arbitrage condition is given by

      $\tilde{\alpha }\left(t,T\right)={\sigma }_1\left(t,T\right){\displaystyle {\int }_t^T{\sigma }_1\left(t,s\right)}\text{d}s+{\sigma }_2\left(t,T\right){\displaystyle {\int }_t^T{\sigma }_2\left(t,s\right)}\text{d}s.$

   3. (c) Provide the multidimensional extension to the formulas in part (a) and (b) for the more general case of a d-factor HJM model for any d ≥ 1.

8. Exercise 15.8. Let the dynamics of the stock price S(t) and the bond price Z(t, T) be respectively governed by (15.105) and (15.106). Show that the dynamics of the forward price $F\left(t\right)=\frac{S\left(t\right)}{Z\left(t,T\right)}$ under the T-forward measure $\widehat{\mathbb{P}}\equiv {\widehat{\mathbb{P}}}_T^{\left(Z\right)}$ is governed by 15.107.

9. Exercise 15.9. Consider the multidimensional extension of (15.105)-(15.106) where each stock price process {Si(t)}0≤t≤T, i = 1,..., n, and the zero-coupon bond price Z(t, T) are all driven by a vector Brownian motion W(t) = (W1(t),..., Wd(t)). Assume the existence of a risk-neutral measure $\tilde{\mathbb{P}}\equiv {\tilde{\mathbb{P}}}^{\left(B\right)}$, i.e.,

   $\begin{array}{l}\frac{\text{d}{S}_i\left(t\right)}{S_i\left(t\right)}=r\left(t\right)\text{d}t+{\sigma }_i\left(t\right)\cdot \text{d}\tilde{W}\left(t\right),\hfill \\ \frac{\text{d}Z\left(t,T\right)}{Z\left(t,T\right)}=r\left(t\right)\text{d}t+{\sigma }_Z\left(t\right)\cdot \text{d}\tilde{W}\left(t\right),\hfill \end{array}$

   where σi(t) and σZ(t) are the respective log-volatility vectors of each stock and bond price. The forward price of each stock is defined by $F_i\left(t\right)=\frac{S_i\left(t\right)}{Z\left(t,T\right)}$.

   1. (a) Obtain the analogue to the SDE in (15.107) for each forward price Fi(t), i = 1,..., n.
   2. (b) Assume a given stock S1(t) ≡ S(t) has constant log-volatility vector sigma;Z(t) = σZ with magnitude ||σS|| = σS and assume a constant vector σZ(t) = σZ with magnitude ||σZ|| = σZ, where σS · σZ = ρσSσZ, ρ ∊ (−1, 1). By employing the measure $\widehat{\mathbb{P}}\equiv {\widehat{\mathbb{P}}}_T^{\left(Z\right)}$, derive a pricing formula for a European call with payoff (S(T) − K)+.
   3. (c) By employing the measure $\widehat{\mathbb{P}}$, obtain a pricing formula for an exchange option with payoff (S2(T) − S1(T))+, assuming constant log-volatility vectors with ||σi|| = σi, σ1 · σ2 = ρσ1σ2, ||σZ|| = σZ, and σZ · σi = ρiσiσZ, ρ ∊ (−1, 1), ρi ∊ (−1, 1), i = 1, 2.
10. Exercise 15.10. Consider the two-factor Vasiček model defined by the two-dimensional system of SDEs in (15.93).

    1. (a) Provide the time-homogeneous PDE for the pricing function V(t, x1, x2) of a zero-coupon bond under this model by directly applying Theorem 11.20 to the conditional expectation

       $V\left(t,x_1,x_2\right)={\tilde{\text{E}}}_{t,x_1,x_2}\left[\text{e}-{\displaystyle {\int }_t^{T}r\left(u,X\left(u\right)\right)}\text{d}u\right].$

       Note that the short rate is given by (15.96). Namely, as a process we have r(t) = r(t, X(t)) = a0 + a1X1(t) + a2X2(t) with spot value r(t, x1, x2) = a0 + a1x1 + a2x2.

    2. (b) Let V(t, x1, x2) = v(τ, x1, x2), τ = T − t, be the solution to the PDE in part (a) where V(t, x1, x2) has the affine form in (15.97), i.e., let

       $\upsilon \left(\tau ,x_1,x_2\right)={\text{e}}^{A\left(\tau \right)+c_1\left(\tau \right)x_1+c_2\left(\tau \right)x_2}$

       for 0 ≤ τ ≤ T. By substituting this form into the PDE of part (a), show that the coefficient functions must satisfy the first order system of ordinary differential equations:

       $\begin{array}{l}{A}^{\prime }\left(\tau \right)=\frac{1}{2}\left(c_1^2\left(\tau \right)+c_2^2\left(\tau \right)\right)-a_0,\hfill \\ {c^{\prime }}_1\left(\tau \right)=-{\beta }_{11}{c}_1\left(\tau \right)-{\beta }_{21}{c}_2\left(\tau \right)-a_1,\hfill \\ {c^{\prime }}_1\left(\tau \right)=-{\beta }_{22}{c}_2\left(\tau \right)-a_2,\hfill \end{array}$

       subject to the initial conditions A(0) = c1(0) = c2(0) = 0.

    3. (c) Solve the system of equations in part (b) and hence obtain the pricing function for the zero-coupon bond.

11. Exercise 15.11. Consider the two-factor CIR model defined by the two-dimensional system of SDEs in (15.94). Repeat parts (a) and (b) of Exercise 15.10.

12. Exercise 15.12. Prove that the forward LIBOR rate volatility v(t, T) and the T- and (T + τ)-maturity bond volatilities σZ(t, T) and σZ(t, T + τ) are related by

    $\upsilon \left(t,T\right)=\left(\frac{1+\tau L\left(t;T,T+\tau \right)}{\tau L\left(t;T,T+\tau \right)}\right)\left[{\sigma }_Z\left(t,T\right)-{\sigma }_Z\left(t,T+\tau \right)\right].$

13. Exercise 15.13. A floorlet is similar to a caplet except that the floating rate is bounded from below. The effective payoff of a floorlet at time T is τ (κ − L(T; T, T + τ))+. Derive the Black pricing formula for a floorlet. Is there a relationship between a floorlet and a caplet?
14. Exercise 15.14. Caps and floors are collections of caplets and floorlets, respectively, applied to periods [Tj, Tj + τ] with j = 1, 2,..., n. Show that a model-independent relationship cap = floor + swap exists.

1 We note the shorthand notation used throughout this chapter where ${\tilde{\text{E}}}_t[\cdot ]\equiv \tilde{\text{E}}\left[\cdot |{ℱ}_t\right]$ is the $\tilde{\mathbb{P}}$-measure expectation conditional on Ft.

2 We remark that the role of the process X(t) with SDE in (11.24) is now played by r(t) with SDE in (15.31) and the dummy variable x in (11.63) is now called r. The variable r is not to be confused with the function denoted by r(t, x) in Theorem 11.9, which is now given by r(t, x) ≡ x, as seen by identifying (11.62) with (15.32) and where φ(x) ≡ 1.

3 If $f:{ℝ}^2\to ℝ$ is a nonrandom (ordinary) function, then ordinary calculus gives the ordinary derivative w.r.t. t as $\frac{\text{d}}{\text{d}t}\left(-{\displaystyle {\int }_t^{T}f\left(t,u\right)\text{d}u}\right)=f\left(t,t\right)-{\displaystyle {\int }_t^T\frac{\partial f}{\partial t}}\left(t,u\right)\text{d}u$ i.e., multiplying both sides by dt, with $\frac{\partial f}{\partial t}\left(t,u\right)\text{d}t=\text{d}f\left(t,u\right)$ for fixed parameter u, gives the ordinary differential $\text{d}\left(-{\displaystyle {\int }_t^{T}f\left(t,u\right)\text{d}u}\right)=f\left(t,t\right)\text{d}t-{\displaystyle {\int }_t^T\text{d}f\left(t,u\right)\text{d}u.\text{If }f\left(t,u\right):\Omega \to ℝ\text{is}\text{an}{ℱ}_t}$-measurable Itô process (for all parameter values u), then this result applies where df(t, u) is a stochastic differential w.r.t. t.